Calculus II - Partial Fractions

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Exponential and Logarithm Functions</a> <ul> <li><a href="/Classes/Alg/ExpFunctions.aspx" class="mm-link">6.1 Exponential Functions</a></li> <li><a href="/Classes/Alg/LogFunctions.aspx" class="mm-link">6.2 Logarithm Functions</a></li> <li><a href="/Classes/Alg/SolveExpEqns.aspx" class="mm-link">6.3 Solving Exponential Equations</a></li> <li><a href="/Classes/Alg/SolveLogEqns.aspx" class="mm-link">6.4 Solving Logarithm Equations</a></li> <li><a href="/Classes/Alg/ExpLogApplications.aspx" class="mm-link">6.5 Applications</a></li> </ul> </li> <li><a href="/Classes/Alg/Systems.aspx" class="mm-link">7. 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Review</a> <ul> <li><a href="/Classes/CalcI/Functions.aspx" class="mm-link">1.1 Functions</a></li> <li><a href="/Classes/CalcI/InverseFunctions.aspx" class="mm-link">1.2 Inverse Functions</a></li> <li><a href="/Classes/CalcI/TrigFcns.aspx" class="mm-link">1.3 Trig Functions</a></li> <li><a href="/Classes/CalcI/TrigEquations.aspx" class="mm-link">1.4 Solving Trig Equations</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" class="mm-link">1.5 Trig Equations with Calculators, Part I</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcII.aspx" class="mm-link">1.6 Trig Equations with Calculators, Part II</a></li> <li><a href="/Classes/CalcI/ExpFunctions.aspx" class="mm-link">1.7 Exponential Functions</a></li> <li><a href="/Classes/CalcI/LogFcns.aspx" class="mm-link">1.8 Logarithm Functions</a></li> <li><a href="/Classes/CalcI/ExpLogEqns.aspx" class="mm-link">1.9 Exponential and Logarithm Equations</a></li> <li><a href="/Classes/CalcI/CommonGraphs.aspx" class="mm-link">1.10 Common Graphs</a></li> </ul> </li> <li><a href="/Classes/CalcI/limitsIntro.aspx" class="mm-link">2. Limits</a> <ul> <li><a href="/Classes/CalcI/Tangents_Rates.aspx" class="mm-link">2.1 Tangent Lines and Rates of Change</a></li> <li><a href="/Classes/CalcI/TheLimit.aspx" class="mm-link">2.2 The Limit</a></li> <li><a href="/Classes/CalcI/OneSidedLimits.aspx" class="mm-link">2.3 One-Sided Limits</a></li> <li><a href="/Classes/CalcI/LimitsProperties.aspx" class="mm-link">2.4 Limit Properties</a></li> <li><a href="/Classes/CalcI/ComputingLimits.aspx" class="mm-link">2.5 Computing Limits</a></li> <li><a href="/Classes/CalcI/InfiniteLimits.aspx" class="mm-link">2.6 Infinite Limits</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityI.aspx" class="mm-link">2.7 Limits At Infinity, Part I</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityII.aspx" class="mm-link">2.8 Limits At Infinity, Part II</a></li> <li><a href="/Classes/CalcI/Continuity.aspx" class="mm-link">2.9 Continuity</a></li> <li><a href="/Classes/CalcI/DefnOfLimit.aspx" class="mm-link">2.10 The Definition of the Limit</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivativeIntro.aspx" class="mm-link">3. Derivatives</a> <ul> <li><a href="/Classes/CalcI/DefnOfDerivative.aspx" class="mm-link">3.1 The Definition of the Derivative</a></li> <li><a href="/Classes/CalcI/DerivativeInterp.aspx" class="mm-link">3.2 Interpretation of the Derivative</a></li> <li><a href="/Classes/CalcI/DiffFormulas.aspx" class="mm-link">3.3 Differentiation Formulas</a></li> <li><a href="/Classes/CalcI/ProductQuotientRule.aspx" class="mm-link">3.4 Product and Quotient Rule</a></li> <li><a href="/Classes/CalcI/DiffTrigFcns.aspx" class="mm-link">3.5 Derivatives of Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffExpLogFcns.aspx" class="mm-link">3.6 Derivatives of Exponential and Logarithm Functions</a></li> <li><a href="/Classes/CalcI/DiffInvTrigFcns.aspx" class="mm-link">3.7 Derivatives of Inverse Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffHyperFcns.aspx" class="mm-link">3.8 Derivatives of Hyperbolic Functions</a></li> <li><a href="/Classes/CalcI/ChainRule.aspx" class="mm-link">3.9 Chain Rule</a></li> <li><a href="/Classes/CalcI/ImplicitDIff.aspx" class="mm-link">3.10 Implicit Differentiation</a></li> <li><a href="/Classes/CalcI/RelatedRates.aspx" class="mm-link">3.11 Related Rates</a></li> <li><a href="/Classes/CalcI/HigherOrderDerivatives.aspx" class="mm-link">3.12 Higher Order Derivatives</a></li> <li><a href="/Classes/CalcI/LogDiff.aspx" class="mm-link">3.13 Logarithmic Differentiation</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivAppsIntro.aspx" class="mm-link">4. Applications of Derivatives</a> <ul> <li><a href="/Classes/CalcI/RateOfChange.aspx" class="mm-link">4.1 Rates of Change</a></li> <li><a href="/Classes/CalcI/CriticalPoints.aspx" class="mm-link">4.2 Critical Points</a></li> <li><a href="/Classes/CalcI/MinMaxValues.aspx" class="mm-link">4.3 Minimum and Maximum Values</a></li> <li><a href="/Classes/CalcI/AbsExtrema.aspx" class="mm-link">4.4 Finding Absolute Extrema</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtI.aspx" class="mm-link">4.5 The Shape of a Graph, Part I</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtII.aspx" class="mm-link">4.6 The Shape of a Graph, Part II</a></li> <li><a href="/Classes/CalcI/MeanValueTheorem.aspx" class="mm-link">4.7 The Mean Value Theorem</a></li> <li><a href="/Classes/CalcI/Optimization.aspx" class="mm-link">4.8 Optimization</a></li> <li><a href="/Classes/CalcI/MoreOptimization.aspx" class="mm-link">4.9 More Optimization Problems</a></li> <li><a href="/Classes/CalcI/LHospitalsRule.aspx" class="mm-link">4.10 L'Hospital's Rule and Indeterminate Forms</a></li> <li><a href="/Classes/CalcI/LinearApproximations.aspx" class="mm-link">4.11 Linear Approximations</a></li> <li><a href="/Classes/CalcI/Differentials.aspx" class="mm-link">4.12 Differentials</a></li> <li><a href="/Classes/CalcI/NewtonsMethod.aspx" class="mm-link">4.13 Newton's Method</a></li> <li><a href="/Classes/CalcI/BusinessApps.aspx" class="mm-link">4.14 Business Applications</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntegralsIntro.aspx" class="mm-link">5. Integrals</a> <ul> <li><a href="/Classes/CalcI/IndefiniteIntegrals.aspx" class="mm-link">5.1 Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/ComputingIndefiniteIntegrals.aspx" class="mm-link">5.2 Computing Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinite.aspx" class="mm-link">5.3 Substitution Rule for Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx" class="mm-link">5.4 More Substitution Rule</a></li> <li><a href="/Classes/CalcI/AreaProblem.aspx" class="mm-link">5.5 Area Problem</a></li> <li><a href="/Classes/CalcI/DefnOfDefiniteIntegral.aspx" class="mm-link">5.6 Definition of the Definite Integral</a></li> <li><a href="/Classes/CalcI/ComputingDefiniteIntegrals.aspx" class="mm-link">5.7 Computing Definite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleDefinite.aspx" class="mm-link">5.8 Substitution Rule for Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntAppsIntro.aspx" class="mm-link">6. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcI/AvgFcnValue.aspx" class="mm-link">6.1 Average Function Value</a></li> <li><a href="/Classes/CalcI/AreaBetweenCurves.aspx" class="mm-link">6.2 Area Between Curves</a></li> <li><a href="/Classes/CalcI/VolumeWithRings.aspx" class="mm-link">6.3 Volumes of Solids of Revolution / Method of Rings</a></li> <li><a href="/Classes/CalcI/VolumeWithCylinder.aspx" class="mm-link">6.4 Volumes of Solids of Revolution/Method of Cylinders</a></li> <li><a href="/Classes/CalcI/MoreVolume.aspx" class="mm-link">6.5 More Volume Problems</a></li> <li><a href="/Classes/CalcI/Work.aspx" class="mm-link">6.6 Work</a></li> </ul> </li> <li><a href="/Classes/CalcI/ExtrasIntro.aspx" class="mm-link">Appendix A. Extras</a> <ul> <li><a href="/Classes/CalcI/LimitProofs.aspx" class="mm-link">A.1 Proof of Various Limit Properties</a></li> <li><a href="/Classes/CalcI/DerivativeProofs.aspx" class="mm-link">A.2 Proof of Various Derivative Properties</a></li> <li><a href="/Classes/CalcI/ProofTrigDeriv.aspx" class="mm-link">A.3 Proof of Trig Limits</a></li> <li><a href="/Classes/CalcI/DerivativeAppsProofs.aspx" class="mm-link">A.4 Proofs of Derivative Applications Facts</a></li> <li><a href="/Classes/CalcI/ProofIntProp.aspx" class="mm-link">A.5 Proof of Various Integral Properties </a></li> <li><a href="/Classes/CalcI/Area_Volume_Formulas.aspx" class="mm-link">A.6 Area and Volume Formulas</a></li> <li><a href="/Classes/CalcI/TypesOfInfinity.aspx" class="mm-link">A.7 Types of Infinity</a></li> <li><a href="/Classes/CalcI/SummationNotation.aspx" class="mm-link">A.8 Summation Notation</a></li> <li><a href="/Classes/CalcI/ConstantofIntegration.aspx" class="mm-link">A.9 Constant of Integration</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcII/CalcII.aspx" class="mm-link">Calculus II</a> <ul> <li><a href="/Classes/CalcII/IntTechIntro.aspx" class="mm-link">7. Integration Techniques</a> <ul> <li><a href="/Classes/CalcII/IntegrationByParts.aspx" class="mm-link">7.1 Integration by Parts</a></li> <li><a href="/Classes/CalcII/IntegralsWithTrig.aspx" class="mm-link">7.2 Integrals Involving Trig Functions</a></li> <li><a href="/Classes/CalcII/TrigSubstitutions.aspx" class="mm-link">7.3 Trig Substitutions</a></li> <li><a href="/Classes/CalcII/PartialFractions.aspx" class="mm-link">7.4 Partial Fractions</a></li> <li><a href="/Classes/CalcII/IntegralsWithRoots.aspx" class="mm-link">7.5 Integrals Involving Roots</a></li> <li><a href="/Classes/CalcII/IntegralsWithQuadratics.aspx" class="mm-link">7.6 Integrals Involving Quadratics</a></li> <li><a href="/Classes/CalcII/IntegrationStrategy.aspx" class="mm-link">7.7 Integration Strategy</a></li> <li><a href="/Classes/CalcII/ImproperIntegrals.aspx" class="mm-link">7.8 Improper Integrals</a></li> <li><a href="/Classes/CalcII/ImproperIntegralsCompTest.aspx" class="mm-link">7.9 Comparison Test for Improper Integrals</a></li> <li><a href="/Classes/CalcII/ApproximatingDefIntegrals.aspx" class="mm-link">7.10 Approximating Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcII/IntAppsIntro.aspx" class="mm-link">8. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcII/ArcLength.aspx" class="mm-link">8.1 Arc Length</a></li> <li><a href="/Classes/CalcII/SurfaceArea.aspx" class="mm-link">8.2 Surface Area</a></li> <li><a href="/Classes/CalcII/CenterOfMass.aspx" class="mm-link">8.3 Center of Mass</a></li> <li><a href="/Classes/CalcII/HydrostaticPressure.aspx" class="mm-link">8.4 Hydrostatic Pressure</a></li> <li><a href="/Classes/CalcII/Probability.aspx" class="mm-link">8.5 Probability</a></li> </ul> </li> <li><a href="/Classes/CalcII/ParametricIntro.aspx" class="mm-link">9. Parametric Equations and Polar Coordinates</a> <ul> <li><a href="/Classes/CalcII/ParametricEqn.aspx" class="mm-link">9.1 Parametric Equations and Curves</a></li> <li><a href="/Classes/CalcII/ParaTangent.aspx" class="mm-link">9.2 Tangents with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArea.aspx" class="mm-link">9.3 Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArcLength.aspx" class="mm-link">9.4 Arc Length with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaSurfaceArea.aspx" class="mm-link">9.5 Surface Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/PolarCoordinates.aspx" class="mm-link">9.6 Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarTangents.aspx" class="mm-link">9.7 Tangents with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArea.aspx" class="mm-link">9.8 Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArcLength.aspx" class="mm-link">9.9 Arc Length with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarSurfaceArea.aspx" class="mm-link">9.10 Surface Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/ArcLength_SurfaceArea.aspx" class="mm-link">9.11 Arc Length and Surface Area Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcII/SeriesIntro.aspx" class="mm-link">10. Series & Sequences</a> <ul> <li><a href="/Classes/CalcII/Sequences.aspx" class="mm-link">10.1 Sequences</a></li> <li><a href="/Classes/CalcII/MoreSequences.aspx" class="mm-link">10.2 More on Sequences</a></li> <li><a href="/Classes/CalcII/Series_Basics.aspx" class="mm-link">10.3 Series - The Basics</a></li> <li><a href="/Classes/CalcII/ConvergenceOfSeries.aspx" class="mm-link">10.4 Convergence/Divergence of Series</a></li> <li><a href="/Classes/CalcII/Series_Special.aspx" class="mm-link">10.5 Special Series</a></li> <li><a href="/Classes/CalcII/IntegralTest.aspx" class="mm-link">10.6 Integral Test</a></li> <li><a href="/Classes/CalcII/SeriesCompTest.aspx" class="mm-link">10.7 Comparison Test/Limit Comparison Test</a></li> <li><a href="/Classes/CalcII/AlternatingSeries.aspx" class="mm-link">10.8 Alternating Series Test</a></li> <li><a href="/Classes/CalcII/AbsoluteConvergence.aspx" class="mm-link">10.9 Absolute Convergence</a></li> <li><a href="/Classes/CalcII/RatioTest.aspx" class="mm-link">10.10 Ratio Test</a></li> <li><a href="/Classes/CalcII/RootTest.aspx" class="mm-link">10.11 Root Test</a></li> <li><a href="/Classes/CalcII/SeriesStrategy.aspx" class="mm-link">10.12 Strategy for Series</a></li> <li><a href="/Classes/CalcII/EstimatingSeries.aspx" class="mm-link">10.13 Estimating the Value of a Series</a></li> <li><a href="/Classes/CalcII/PowerSeries.aspx" class="mm-link">10.14 Power Series</a></li> <li><a href="/Classes/CalcII/PowerSeriesandFunctions.aspx" class="mm-link">10.15 Power Series and Functions</a></li> <li><a href="/Classes/CalcII/TaylorSeries.aspx" class="mm-link">10.16 Taylor Series</a></li> <li><a href="/Classes/CalcII/TaylorSeriesApps.aspx" class="mm-link">10.17 Applications of Series</a></li> <li><a href="/Classes/CalcII/BinomialSeries.aspx" class="mm-link">10.18 Binomial Series</a></li> </ul> </li> <li><a href="/Classes/CalcII/VectorsIntro.aspx" class="mm-link">11. Vectors</a> <ul> <li><a href="/Classes/CalcII/Vectors_Basics.aspx" class="mm-link">11.1 Vectors - The Basics</a></li> <li><a href="/Classes/CalcII/VectorArithmetic.aspx" class="mm-link">11.2 Vector Arithmetic</a></li> <li><a href="/Classes/CalcII/DotProduct.aspx" class="mm-link">11.3 Dot Product</a></li> <li><a href="/Classes/CalcII/CrossProduct.aspx" class="mm-link">11.4 Cross Product</a></li> </ul> </li> <li><a href="/Classes/CalcII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcIII/CalcIII.aspx" class="mm-link">Calculus III</a> <ul> <li><a href="/Classes/CalcIII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcIII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcIII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcIII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcIII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcIII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcIII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcIII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcIII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcIII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcIII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcIII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcIII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivsIntro.aspx" class="mm-link">13. Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/Limits.aspx" class="mm-link">13.1 Limits</a></li> <li><a href="/Classes/CalcIII/PartialDerivatives.aspx" class="mm-link">13.2 Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/PartialDerivInterp.aspx" class="mm-link">13.3 Interpretations of Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/HighOrderPartialDerivs.aspx" class="mm-link">13.4 Higher Order Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/Differentials.aspx" class="mm-link">13.5 Differentials</a></li> <li><a href="/Classes/CalcIII/ChainRule.aspx" class="mm-link">13.6 Chain Rule</a></li> <li><a href="/Classes/CalcIII/DirectionalDeriv.aspx" class="mm-link">13.7 Directional Derivatives</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivAppsIntro.aspx" class="mm-link">14. Applications of Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/TangentPlanes.aspx" class="mm-link">14.1 Tangent Planes and Linear Approximations</a></li> <li><a href="/Classes/CalcIII/GradientVectorTangentPlane.aspx" class="mm-link">14.2 Gradient Vector, Tangent Planes and Normal Lines</a></li> <li><a href="/Classes/CalcIII/RelativeExtrema.aspx" class="mm-link">14.3 Relative Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/AbsoluteExtrema.aspx" class="mm-link">14.4 Absolute Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/LagrangeMultipliers.aspx" class="mm-link">14.5 Lagrange Multipliers</a></li> </ul> </li> <li><a href="/Classes/CalcIII/MultipleIntegralsIntro.aspx" class="mm-link">15. Multiple Integrals</a> <ul> <li><a href="/Classes/CalcIII/DoubleIntegrals.aspx" class="mm-link">15.1 Double Integrals</a></li> <li><a href="/Classes/CalcIII/IteratedIntegrals.aspx" class="mm-link">15.2 Iterated Integrals</a></li> <li><a href="/Classes/CalcIII/DIGeneralRegion.aspx" class="mm-link">15.3 Double Integrals over General Regions</a></li> <li><a href="/Classes/CalcIII/DIPolarCoords.aspx" class="mm-link">15.4 Double Integrals in Polar Coordinates</a></li> <li><a href="/Classes/CalcIII/TripleIntegrals.aspx" class="mm-link">15.5 Triple Integrals</a></li> <li><a href="/Classes/CalcIII/TICylindricalCoords.aspx" class="mm-link">15.6 Triple Integrals in Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/TISphericalCoords.aspx" class="mm-link">15.7 Triple Integrals in Spherical Coordinates</a></li> <li><a href="/Classes/CalcIII/ChangeOfVariables.aspx" class="mm-link">15.8 Change of Variables</a></li> <li><a href="/Classes/CalcIII/SurfaceArea.aspx" class="mm-link">15.9 Surface Area</a></li> <li><a href="/Classes/CalcIII/Area_Volume.aspx" class="mm-link">15.10 Area and Volume Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" class="mm-link">16. Line Integrals</a> <ul> <li><a href="/Classes/CalcIII/VectorFields.aspx" class="mm-link">16.1 Vector Fields</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtI.aspx" class="mm-link">16.2 Line Integrals - Part I</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtII.aspx" class="mm-link">16.3 Line Integrals - Part II</a></li> <li><a href="/Classes/CalcIII/LineIntegralsVectorFields.aspx" class="mm-link">16.4 Line Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/FundThmLineIntegrals.aspx" class="mm-link">16.5 Fundamental Theorem for Line Integrals</a></li> <li><a href="/Classes/CalcIII/ConservativeVectorField.aspx" class="mm-link">16.6 Conservative Vector Fields</a></li> <li><a href="/Classes/CalcIII/GreensTheorem.aspx" class="mm-link">16.7 Green's Theorem</a></li> </ul> </li> <li><a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" class="mm-link">17.Surface Integrals</a> <ul> <li><a href="/Classes/CalcIII/CurlDivergence.aspx" class="mm-link">17.1 Curl and Divergence</a></li> <li><a href="/Classes/CalcIII/ParametricSurfaces.aspx" class="mm-link">17.2 Parametric Surfaces</a></li> <li><a href="/Classes/CalcIII/SurfaceIntegrals.aspx" class="mm-link">17.3 Surface Integrals</a></li> <li><a href="/Classes/CalcIII/SurfIntVectorField.aspx" class="mm-link">17.4 Surface Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/StokesTheorem.aspx" class="mm-link">17.5 Stokes' Theorem</a></li> <li><a href="/Classes/CalcIII/DivergenceTheorem.aspx" class="mm-link">17.6 Divergence Theorem</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/DE/DE.aspx" class="mm-link">Differential Equations</a> <ul> <li><a href="/Classes/DE/IntroBasic.aspx" class="mm-link">1. Basic Concepts</a> <ul> <li><a href="/Classes/DE/Definitions.aspx" class="mm-link">1.1 Definitions</a></li> <li><a href="/Classes/DE/DirectionFields.aspx" class="mm-link">1.2 Direction Fields</a></li> <li><a href="/Classes/DE/FinalThoughts.aspx" class="mm-link">1.3 Final Thoughts</a></li> </ul> </li> <li><a href="/Classes/DE/IntroFirstOrder.aspx" class="mm-link">2. First Order DE's</a> <ul> <li><a href="/Classes/DE/Linear.aspx" class="mm-link">2.1 Linear Equations</a></li> <li><a href="/Classes/DE/Separable.aspx" class="mm-link">2.2 Separable Equations</a></li> <li><a href="/Classes/DE/Exact.aspx" class="mm-link">2.3 Exact Equations</a></li> <li><a href="/Classes/DE/Bernoulli.aspx" class="mm-link">2.4 Bernoulli Differential Equations</a></li> <li><a href="/Classes/DE/Substitutions.aspx" class="mm-link">2.5 Substitutions</a></li> <li><a href="/Classes/DE/IoV.aspx" class="mm-link">2.6 Intervals of Validity</a></li> <li><a href="/Classes/DE/Modeling.aspx" class="mm-link">2.7 Modeling with First Order DE's</a></li> <li><a href="/Classes/DE/EquilibriumSolutions.aspx" class="mm-link">2.8 Equilibrium Solutions</a></li> <li><a href="/Classes/DE/EulersMethod.aspx" class="mm-link">2.9 Euler's Method</a></li> </ul> </li> <li><a href="/Classes/DE/IntroSecondOrder.aspx" class="mm-link">3. Second Order DE's</a> <ul> <li><a href="/Classes/DE/SecondOrderConcepts.aspx" class="mm-link">3.1 Basic Concepts</a></li> <li><a href="/Classes/DE/RealRoots.aspx" class="mm-link">3.2 Real & Distinct Roots</a></li> <li><a href="/Classes/DE/ComplexRoots.aspx" class="mm-link">3.3 Complex Roots</a></li> <li><a href="/Classes/DE/RepeatedRoots.aspx" class="mm-link">3.4 Repeated Roots</a></li> <li><a href="/Classes/DE/ReductionofOrder.aspx" class="mm-link">3.5 Reduction of Order</a></li> <li><a href="/Classes/DE/FundamentalSetsofSolutions.aspx" class="mm-link">3.6 Fundamental Sets of Solutions</a></li> <li><a href="/Classes/DE/Wronskian.aspx" class="mm-link">3.7 More on the Wronskian</a></li> <li><a href="/Classes/DE/NonhomogeneousDE.aspx" class="mm-link">3.8 Nonhomogeneous Differential Equations</a></li> <li><a href="/Classes/DE/UndeterminedCoefficients.aspx" class="mm-link">3.9 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/VariationofParameters.aspx" class="mm-link">3.10 Variation of Parameters</a></li> <li><a href="/Classes/DE/Vibrations.aspx" class="mm-link">3.11 Mechanical Vibrations</a></li> </ul> </li> <li><a href="/Classes/DE/LaplaceIntro.aspx" class="mm-link">4. Laplace Transforms</a> <ul> <li><a href="/Classes/DE/LaplaceDefinition.aspx" class="mm-link">4.1 The Definition</a></li> <li><a href="/Classes/DE/LaplaceTransforms.aspx" class="mm-link">4.2 Laplace Transforms</a></li> <li><a href="/Classes/DE/InverseTransforms.aspx" class="mm-link">4.3 Inverse Laplace Transforms</a></li> <li><a href="/Classes/DE/StepFunctions.aspx" class="mm-link">4.4 Step Functions</a></li> <li><a href="/Classes/DE/IVPWithLaplace.aspx" class="mm-link">4.5 Solving IVP's with Laplace Transforms</a></li> <li><a href="/Classes/DE/IVPWithNonConstantCoefficient.aspx" class="mm-link">4.6 Nonconstant Coefficient IVP's</a></li> <li><a href="/Classes/DE/IVPWithStepFunction.aspx" class="mm-link">4.7 IVP's With Step Functions</a></li> <li><a href="/Classes/DE/DiracDeltaFunction.aspx" class="mm-link">4.8 Dirac Delta Function</a></li> <li><a href="/Classes/DE/ConvolutionIntegrals.aspx" class="mm-link">4.9 Convolution Integrals</a></li> <li><a href="/Classes/DE/Laplace_Table.aspx" class="mm-link">4.10 Table Of Laplace Transforms</a></li> </ul> </li> <li><a href="/Classes/DE/SystemsIntro.aspx" class="mm-link">5. Systems of DE's</a> <ul> <li><a href="/Classes/DE/LA_Systems.aspx" class="mm-link">5.1 Review : Systems of Equations</a></li> <li><a href="/Classes/DE/LA_Matrix.aspx" class="mm-link">5.2 Review : Matrices & Vectors</a></li> <li><a href="/Classes/DE/LA_Eigen.aspx" class="mm-link">5.3 Review : Eigenvalues & Eigenvectors</a></li> <li><a href="/Classes/DE/SystemsDE.aspx" class="mm-link">5.4 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/SolutionsToSystems.aspx" class="mm-link">5.5 Solutions to Systems</a></li> <li><a href="/Classes/DE/PhasePlane.aspx" class="mm-link">5.6 Phase Plane</a></li> <li><a href="/Classes/DE/RealEigenvalues.aspx" class="mm-link">5.7 Real Eigenvalues</a></li> <li><a href="/Classes/DE/ComplexEigenvalues.aspx" class="mm-link">5.8 Complex Eigenvalues</a></li> <li><a href="/Classes/DE/RepeatedEigenvalues.aspx" class="mm-link">5.9 Repeated Eigenvalues</a></li> <li><a href="/Classes/DE/NonhomogeneousSystems.aspx" class="mm-link">5.10 Nonhomogeneous Systems</a></li> <li><a href="/Classes/DE/SystemsLaplace.aspx" class="mm-link">5.11 Laplace Transforms</a></li> <li><a href="/Classes/DE/SystemsModeling.aspx" class="mm-link">5.12 Modeling</a></li> </ul> </li> <li><a href="/Classes/DE/SeriesIntro.aspx" class="mm-link">6. Series Solutions to DE's</a> <ul> <li><a href="/Classes/DE/PowerSeries.aspx" class="mm-link">6.1 Review : Power Series</a></li> <li><a href="/Classes/DE/TaylorSeries.aspx" class="mm-link">6.2 Review : Taylor Series</a></li> <li><a href="/Classes/DE/SeriesSolutions.aspx" class="mm-link">6.3 Series Solutions</a></li> <li><a href="/Classes/DE/EulerEquations.aspx" class="mm-link">6.4 Euler Equations</a></li> </ul> </li> <li><a href="/Classes/DE/IntroHigherOrder.aspx" class="mm-link">7. Higher Order Differential Equations</a> <ul> <li><a href="/Classes/DE/HOBasicConcepts.aspx" class="mm-link">7.1 Basic Concepts for <em>n</em><sup>th</sup> Order Linear Equations</a></li> <li><a href="/Classes/DE/HOHomogeneousDE.aspx" class="mm-link">7.2 Linear Homogeneous Differential Equations</a></li> <li><a href="/Classes/DE/HOUndeterminedCoeff.aspx" class="mm-link">7.3 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/HOVariationOfParam.aspx" class="mm-link">7.4 Variation of Parameters</a></li> <li><a href="/Classes/DE/HOLaplaceTransforms.aspx" class="mm-link">7.5 Laplace Transforms</a></li> <li><a href="/Classes/DE/HOSystems.aspx" class="mm-link">7.6 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/HOSeries.aspx" class="mm-link">7.7 Series Solutions</a></li> </ul> </li> <li><a href="/Classes/DE/IntroBVP.aspx" class="mm-link">8. Boundary Value Problems & Fourier Series</a> <ul> <li><a href="/Classes/DE/BoundaryValueProblem.aspx" class="mm-link">8.1 Boundary Value Problems</a></li> <li><a href="/Classes/DE/BVPEvals.aspx" class="mm-link">8.2 Eigenvalues and Eigenfunctions</a></li> <li><a href="/Classes/DE/PeriodicOrthogonal.aspx" class="mm-link">8.3 Periodic Functions & Orthogonal Functions</a></li> <li><a href="/Classes/DE/FourierSineSeries.aspx" class="mm-link">8.4 Fourier Sine Series</a></li> <li><a href="/Classes/DE/FourierCosineSeries.aspx" class="mm-link">8.5 Fourier Cosine Series</a></li> <li><a href="/Classes/DE/FourierSeries.aspx" class="mm-link">8.6 Fourier Series</a></li> <li><a href="/Classes/DE/ConvergenceFourierSeries.aspx" class="mm-link">8.7 Convergence of Fourier Series</a></li> </ul> </li> <li><a href="/Classes/DE/IntroPDE.aspx" class="mm-link">9. Partial Differential Equations </a> <ul> <li><a href="/Classes/DE/TheHeatEquation.aspx" class="mm-link">9.1 The Heat Equation</a></li> <li><a href="/Classes/DE/TheWaveEquation.aspx" class="mm-link">9.2 The Wave Equation</a></li> <li><a href="/Classes/DE/PDETerminology.aspx" class="mm-link">9.3 Terminology</a></li> <li><a href="/Classes/DE/SeparationofVariables.aspx" class="mm-link">9.4 Separation of Variables</a></li> <li><a href="/Classes/DE/SolvingHeatEquation.aspx" class="mm-link">9.5 Solving the Heat Equation</a></li> <li><a href="/Classes/DE/HeatEqnNonZero.aspx" class="mm-link">9.6 Heat Equation with Non-Zero Temperature Boundaries</a></li> <li><a href="/Classes/DE/LaplacesEqn.aspx" class="mm-link">9.7 Laplace's Equation</a></li> <li><a href="/Classes/DE/VibratingString.aspx" class="mm-link">9.8 Vibrating String</a></li> <li><a href="/Classes/DE/PDESummary.aspx" class="mm-link">9.9 Summary of Separation of Variables</a></li> </ul> </li> </ul> </li> <li><span>Extras</span></li> <li><a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" class="mm-link">Algebra & Trig Review</a> <ul> <li><a href="/Extras/AlgebraTrigReview/AlgebraIntro.aspx" class="mm-link">1. Algebra</a> <ul> <li><a href="/Extras/AlgebraTrigReview/Exponents.aspx" class="mm-link">1.1 Exponents </a></li> <li><a href="/Extras/AlgebraTrigReview/AbsoluteValue.aspx" class="mm-link">1.2 Absolute Value</a></li> <li><a href="/Extras/AlgebraTrigReview/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Extras/AlgebraTrigReview/Rationalizing.aspx" class="mm-link">1.4 Rationalizing </a></li> <li><a href="/Extras/AlgebraTrigReview/Functions.aspx" class="mm-link">1.5 Functions </a></li> <li><a href="/Extras/AlgebraTrigReview/MultPoly.aspx" class="mm-link">1.6 Multiplying Polynomials</a></li> <li><a href="/Extras/AlgebraTrigReview/Factoring.aspx" class="mm-link">1.7 Factoring</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpRatExp.aspx" class="mm-link">1.8 Simplifying Rational Expressions</a></li> <li><a href="/Extras/AlgebraTrigReview/Graphing.aspx" class="mm-link">1.9 Graphing and Common Graphs</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtI.aspx" class="mm-link">1.10 Solving Equations, Part I</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtII.aspx" class="mm-link">1.11 Solving Equations, Part II</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveSystems.aspx" class="mm-link">1.12 Solving Systems of Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveIneq.aspx" class="mm-link">1.13 Solving Inequalities</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveAbsValue.aspx" class="mm-link">1.14 Absolute Value Equations and Inequalities</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/TrigIntro.aspx" class="mm-link">2. Trigonometry</a> <ul> <li><a href="/Extras/AlgebraTrigReview/TrigFunctions.aspx" class="mm-link">2.1 Trig Function Evaluation</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigGraphs.aspx" class="mm-link">2.2 Graphs of Trig Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigFormulas.aspx" class="mm-link">2.3 Trig Formulas</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveTrigEqn.aspx" class="mm-link">2.4 Solving Trig Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/InverseTrig.aspx" class="mm-link">2.5 Inverse Trig Functions</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/ExpLogIntro.aspx" class="mm-link">3. Exponentials & Logarithms</a> <ul> <li><a href="/Extras/AlgebraTrigReview/ExponentialFcns.aspx" class="mm-link">3.1 Basic Exponential Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogarithmFcns.aspx" class="mm-link">3.2 Basic Logarithm Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogProperties.aspx" class="mm-link">3.3 Logarithm Properties</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpLogs.aspx" class="mm-link">3.4 Simplifying Logarithms</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveExpEqn.aspx" class="mm-link">3.5 Solving Exponential Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveLogEqn.aspx" class="mm-link">3.6 Solving Logarithm Equations</a></li> </ul> </li> </ul> </li> <li><a href="/Extras/CommonErrors/CommonMathErrors.aspx" class="mm-link">Common Math Errors</a> <ul> <li><a href="/Extras/CommonErrors/GeneralErrors.aspx" class="mm-link">1. General Errors</a> </li> <li><a href="/Extras/CommonErrors/AlgebraErrors.aspx" class="mm-link">2. Algebra Errors</a> </li> <li><a href="/Extras/CommonErrors/TrigErrors.aspx" class="mm-link">3. Trig Errors</a> </li> <li><a href="/Extras/CommonErrors/CommonErrors.aspx" class="mm-link">4. Common Errors</a> </li> <li><a href="/Extras/CommonErrors/CalculusErrors.aspx" class="mm-link">5. Calculus Errors</a> </li> </ul> </li> <li><a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" class="mm-link">Complex Number Primer</a> <ul> <li><a href="/Extras/ComplexPrimer/Definition.aspx" class="mm-link">1. The Definition</a> </li> <li><a href="/Extras/ComplexPrimer/Arithmetic.aspx" class="mm-link">2. Arithmetic</a> </li> <li><a href="/Extras/ComplexPrimer/ConjugateModulus.aspx" class="mm-link">3. Conjugate and Modulus</a> </li> <li><a href="/Extras/ComplexPrimer/Forms.aspx" class="mm-link">4. Polar and Exponential Forms</a> </li> <li><a href="/Extras/ComplexPrimer/Roots.aspx" class="mm-link">5. 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However, often the numerator isn鈥檛 the derivative of the denominator (or a constant multiple). For example, consider the following integral.</p> \[\int{{\frac{{3x + 11}}{{{x^2} - x - 6}}\,dx}}\] <p>In this case the numerator is definitely not the derivative of the denominator nor is it a constant multiple of the derivative of the denominator. Therefore, the simple substitution that we used above won鈥檛 work. However, if we notice that the integrand can be broken up as follows,</p> \[\frac{{3x + 11}}{{{x^2} - x - 6}} = \frac{4}{{x - 3}} - \frac{1}{{x + 2}}\] <p>then the integral is actually quite simple.</p> \[\begin{align*}\int{{\frac{{3x + 11}}{{{x^2} - x - 6}}\,dx}} & = \int{{\frac{4}{{x - 3}}\, - \frac{1}{{x + 2}}dx}}\\ & = 4\ln \left| {x - 3} \right| - \ln \left| {x + 2} \right| + c\end{align*}\] <p>This process of taking a rational expression and decomposing it into simpler rational expressions that we can add or subtract to get the original rational expression is called <strong>partial fraction decomposition</strong>. Many integrals involving rational expressions can be done if we first do partial fractions on the integrand.</p> <p>So, let鈥檚 do a quick review of partial fractions. We鈥檒l start with a rational expression in the form,</p> \[f\left( x \right) = \frac{{P\left( x \right)}}{{Q\left( x \right)}}\] <p>where both $P\left( x \right)$ and $Q\left( x \right)$ are polynomials and the degree of $P\left( x \right)$ is smaller than the degree of $Q\left( x \right)$. Recall that the degree of a polynomial is the largest exponent in the polynomial. Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator. That is important to remember.</p> <p>So, once we鈥檝e determined that partial fractions can be done we factor the denominator as completely as possible. Then for each factor in the denominator we can use the following table to determine the term(s) we pick up in the partial fraction decomposition.</p> <table class="table-no-outer-border"> <tr> <th>Factor in<br />denominator</th> <th style="border-right:unset">Term in partial<br />fraction decomposition</th> </tr> <tr> <td class="table-cell-center">$ \displaystyle ax + b$</td> <td class="table-cell-center" style="border-right:unset">$ \displaystyle \frac{A}{{ax + b}}$</td> </tr> <tr> <td class="table-cell-center">$ \displaystyle {\left( {ax + b} \right)^k}$</td> <td class="table-cell-center" style="border-right:unset">$ \displaystyle \frac{{{A_1}}}{{ax + b}} + \frac{{{A_2}}}{{{{\left( {ax + b} \right)}^2}}} + \cdots + \frac{{{A_k}}}{{{{\left( {ax + b} \right)}^k}}}$, $k = 1,2,3, \ldots $</p> </td> </tr> <tr> <td class="table-cell-center">$ \displaystyle a{x^2} + bx + c$</td> <td class="table-cell-center" style="border-right:unset">$ \displaystyle \frac{{Ax + B}}{{a{x^2} + bx + c}}$</td> </tr> <tr> <td class="table-cell-center">$ \displaystyle {\left( {a{x^2} + bx + c} \right)^k}$</td> <td class="table-cell-center" style="border-right:unset">$ \displaystyle \frac{{{A_1}x + {B_1}}}{{a{x^2} + bx + c}} + \frac{{{A_2}x + {B_2}}}{{{{\left( {a{x^2} + bx + c} \right)}^2}}} + \cdots + \frac{{{A_k}x + {B_k}}}{{{{\left( {a{x^2} + bx + c} \right)}^k}}}$, $k = 1,2,3, \ldots $</td> </tr> </table> <p>Notice that the first and third cases are really special cases of the second and fourth cases respectively.</p> <p>There are several methods for determining the coefficients for each term and we will go over each of those in the following examples.</p> <p>Let鈥檚 start the examples by doing the integral above.</p> <a class="anchor" name="Int_PartFrac_Ex1"></a> <div class="example"> <span class="example-title">Example 1</span> Evaluate the following integral. \[\int{{\frac{{3x + 11}}{{{x^2} - x - 6}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln1" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln1" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln1" class="soln-content"> <p>The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this gives,</p> \[\frac{{3x + 11}}{{\left( {x - 3} \right)\left( {x + 2} \right)}}\, = \frac{A}{{x - 3}} + \frac{B}{{x + 2}}\] <p>The next step is to actually add the right side back up.</p> \[\frac{{3x + 11}}{{\left( {x - 3} \right)\left( {x + 2} \right)}}\, = \frac{{A\left( {x + 2} \right) + B\left( {x - 3} \right)}}{{\left( {x - 3} \right)\left( {x + 2} \right)}}\] <p>Now, we need to choose $A$ and $B$ so that the numerators of these two are equal for every $x$. To do this we鈥檒l need to set the numerators equal.</p> \[3x + 11 = A\left( {x + 2} \right) + B\left( {x - 3} \right)\] <p>Note that in most problems we will go straight from the general form of the decomposition to this step and not bother with actually adding the terms back up. The only point to adding the terms is to get the numerator and we can get that without actually writing down the results of the addition.</p> <p>At this point we have one of two ways to proceed. One way will always work but is often more work. The other, while it won鈥檛 always work, is often quicker when it does work. In this case both will work and so we鈥檒l use the quicker way for this example. We鈥檒l take a look at the other method in a later example.</p> <p>What we鈥檙e going to do here is to notice that the numerators must be equal for <em>any x</em> that we would choose to use. In particular the numerators must be equal for $x = - 2$ and $x = 3$. So, let鈥檚 plug these in and see what we get.</p> \[\begin{align*}x & = - 2 : & \hspace{0.5in}5 & = A\left( 0 \right) + B\left( { - 5} \right) & \hspace{0.25in} & \Rightarrow & \hspace{0.25in}B & = - 1\\ x & = 3 \,\,\,\,: & \hspace{0.5in}20 & = A\left( 5 \right) + B\left( 0 \right) & \hspace{0.25in} & \Rightarrow & \hspace{0.25in}A & = 4\end{align*}\] <p>So, by carefully picking the $x$鈥檚 we got the unknown constants to quickly drop out. Note that these are the values we claimed they would be above.</p> <p>At this point there really isn鈥檛 a whole lot to do other than the integral.</p> \[\begin{align*}\int{{\frac{{3x + 11}}{{{x^2} - x - 6}}\,dx}} & = \int{{\frac{4}{{x - 3}}\, - \frac{1}{{x + 2}}dx}}\\ & = \int{{\frac{4}{{x - 3}}\,dx}} - \int{{\frac{1}{{x + 2}}dx}}\\ & = 4\ln \left| {x - 3} \right| - \ln \left| {x + 2} \right| + c\end{align*}\] <p>Recall that to do this integral we first split it up into two integrals and then used the substitutions,</p> \[u = x - 3\hspace{0.5in}v = x + 2\] <p>on the integrals to get the final answer.</p> </div> </div> </div> <p>Before moving onto the next example a couple of quick notes are in order here. First, many of the integrals in partial fractions problems come down to the type of integral seen above. Make sure that you can do those integrals.</p> <p>There is also another integral that often shows up in these kinds of problems so we may as well give the formula for it here since we are already on the subject.</p> \[\int{{\frac{1}{{{x^2} + {a^2}}}\,dx}} = \frac{1}{a}{\tan ^{ - 1}}\left( {\frac{x}{a}} \right) + c\] <p>It will be an example or two before we use this so don鈥檛 forget about it.</p> <p>Now, let鈥檚 work some more examples.</p> <a class="anchor" name="Int_PartFrac_Ex2"></a> <div class="example"> <span class="example-title">Example 2</span> Evaluate the following integral. \[\int{{\frac{{{x^2} + 4}}{{3{x^3} + 4{x^2} - 4x}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln2" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2" class="soln-content"> <p>We won鈥檛 be putting as much detail into this solution as we did in the previous example. The first thing is to factor the denominator and get the form of the partial fraction decomposition.</p> \[\frac{{{x^2} + 4}}{{x\left( {x + 2} \right)\left( {3x - 2} \right)}}\, = \frac{A}{x} + \frac{B}{{x + 2}} + \frac{C}{{3x - 2}}\] <p>The next step is to set numerators equal. If you need to actually add the right side together to get the numerator for that side then you should do so, however, it will definitely make the problem quicker if you can do the addition in your head to get,</p> \[{x^2} + 4 = A\left( {x + 2} \right)\left( {3x - 2} \right) + Bx\left( {3x - 2} \right) + Cx\left( {x + 2} \right)\] <p>As with the previous example it looks like we can just pick a few values of $x$ and find the constants so let鈥檚 do that.</p> \[\begin{align*}x & = 0 \,\,\,\,\, : & \hspace{0.5in}4 & = A\left( 2 \right)\left( { - 2} \right) & \hspace{0.5in} & \Rightarrow & \hspace{0.25in}A & = - 1\\ x & = - 2 : & \hspace{0.5in}8 & = B\left( { - 2} \right)\left( { - 8} \right) & \hspace{0.25in}&\Rightarrow & \hspace{0.25in}B & = \frac{1}{2}\\ x & = \frac{2}{3}\,\, : & \hspace{0.5in}\frac{{40}}{9} & = C\left( {\frac{2}{3}} \right)\left( {\frac{8}{3}} \right) & \hspace{0.25in} & \Rightarrow & \hspace{0.25in}C & = \frac{{40}}{{16}} = \frac{5}{2}\end{align*}\] <p>Note that unlike the first example most of the coefficients here are fractions. That is not unusual so don鈥檛 get excited about it when it happens.</p> <p>Now, let鈥檚 do the integral.</p> \[\begin{align*}\int{{\frac{{{x^2} + 4}}{{3{x^3} + 4{x^2} - 4x}}\,dx}} & = \int{{ - \frac{1}{x} + \frac{{\frac{1}{2}}}{{x + 2}} + \frac{{\frac{5}{2}}}{{3x - 2}}\,dx}}\\ & = - \ln \left| x \right| + \frac{1}{2}\ln \left| {x + 2} \right| + \frac{5}{6}\ln \left| {3x - 2} \right| + c\end{align*}\] <p>Again, as noted above, integrals that generate natural logarithms are very common in these problems so make sure you can do them. Also, you were able to correctly do the last integral right? The coefficient of $\frac{5}{6}$ is correct. Make sure that you do the substitution required for the term properly.</p> </div> </div> </div> <a class="anchor" name="Int_PartFrac_Ex3"></a> <div class="example"> <span class="example-title">Example 3</span> Evaluate the following integral. \[\int{{\frac{{{x^2} - 29x + 5}}{{{{\left( {x - 4} \right)}^2}\left( {{x^2} + 3} \right)}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln3" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln3" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln3" class="soln-content"> <p>This time the denominator is already factored so let鈥檚 just jump right to the partial fraction decomposition.</p> \[\frac{{{x^2} - 29x + 5}}{{{{\left( {x - 4} \right)}^2}\left( {{x^2} + 3} \right)}}\, = \frac{A}{{x - 4}} + \frac{B}{{{{\left( {x - 4} \right)}^2}}} + \frac{{Cx + D}}{{{x^2} + 3}}\] <p>Setting numerators gives,</p> \[{x^2} - 29x + 5 = A\left( {x - 4} \right)\left( {{x^2} + 3} \right) + B\left( {{x^2} + 3} \right) + \left( {Cx + D} \right){\left( {x - 4} \right)^2}\] <p>In this case we aren鈥檛 going to be able to just pick values of $x$ that will give us all the constants. Therefore, we will need to work this the second (and often longer) way. The first step is to multiply out the right side and collect all the like terms together. Doing this gives,</p> \[{x^2} - 29x + 5 = \left( {A + C} \right){x^3} + \left( { - 4A + B - 8C + D} \right){x^2} + \left( {3A + 16C - 8D} \right)x - 12A + 3B + 16D\] <p>Now we need to choose $A$, $B$, $C$, and $D$ so that these two are equal. In other words, we will need to set the coefficients of like powers of $x$ equal. This will give a system of equations that can be solved.</p> \[\left. \begin{align*}{x^3} & :\hspace{0.25in} & A + C & = 0\\ {x^2} & :\hspace{0.25in} & - 4A + B - 8C + D & = 1\\ {x^1} & :\hspace{0.25in} & 3A + 16C - 8D & = - 29\\ {x^0} & :\hspace{0.25in} & - 12A + 3B + 16D & = 5\end{align*} \right\}\hspace{0.25in} \Rightarrow \hspace{0.25in}A = 1,\,B = - 5,\,C = - 1,\,D = 2\] <p>Note that we used ${x^0}$ to represent the constants. Also note that these systems can often be quite large and have a fair amount of work involved in solving them. The best way to deal with these is to use some form of computer aided solving techniques.</p> <p>Now, let鈥檚 take a look at the integral.</p> <p></p> \[\begin{align*}\int{{\frac{{{x^2} - 29x + 5}}{{{{\left( {x - 4} \right)}^2}\left( {{x^2} + 3} \right)}}\,dx}} & = \int{{\frac{1}{{x - 4}} - \frac{5}{{{{\left( {x - 4} \right)}^2}}} + \frac{{ - x + 2}}{{{x^2} + 3}}\,dx}}\\ & = \int{{\frac{1}{{x - 4}} - \frac{5}{{{{\left( {x - 4} \right)}^2}}} - \frac{x}{{{x^2} + 3}}\, + \frac{2}{{{x^2} + 3}}\,dx}}\\ & = \ln \left| {x - 4} \right| + \frac{5}{{x - 4}} - \frac{1}{2}\ln \left| {{x^2} + 3} \right| + \frac{2}{{\sqrt 3 }}{\tan ^{ - 1}}\left( {\frac{x}{{\sqrt 3 }}} \right) + c\end{align*}\] <p>In order to take care of the third term we needed to split it up into two separate terms. Once we鈥檝e done this we can do all the integrals in the problem. The first two use the substitution $u = x - 4$, the third uses the substitution $v = {x^2} + 3$ and the fourth term uses the formula given above for inverse tangents.</p> </div> </div> </div> <a class="anchor" name="Int_PartFrac_Ex4"></a> <div class="example"> <span class="example-title">Example 4</span> Evaluate the following integral. \[\int{{\frac{{{x^3} + 10{x^2} + 3x + 36}}{{\left( {x - 1} \right){{\left( {{x^2} + 4} \right)}^2}}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln4" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln4" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln4" class="soln-content"> <p>Let鈥檚 first get the general form of the partial fraction decomposition.</p> \[\frac{{{x^3} + 10{x^2} + 3x + 36}}{{\left( {x - 1} \right){{\left( {{x^2} + 4} \right)}^2}}}\, = \frac{A}{{x - 1}} + \frac{{Bx + C}}{{{x^2} + 4}} + \frac{{Dx + E}}{{{{\left( {{x^2} + 4} \right)}^2}}}\] <p>Now, set numerators equal, expand the right side and collect like terms.</p> \[\begin{align*}{x^3} + 10{x^2} + 3x + 36 & = A{\left( {{x^2} + 4} \right)^2} + \left( {Bx + C} \right)\left( {x - 1} \right)\left( {{x^2} + 4} \right) + \left( {Dx + E} \right)\left( {x - 1} \right)\\ & = \left( {A + B} \right){x^4} + \left( {C - B} \right){x^3} + \left( {8A + 4B - C + D} \right){x^2} + \\ & \hspace{0.5in}\hspace{0.25in}\left( { - 4B + 4C - D + E} \right)x + 16A - 4C - E\end{align*}\] <p>Setting coefficient equal gives the following system.</p> \[\left. \begin{align*}{x^4} & :\hspace{0.25in} & A + B & = 0\\ {x^3} & :\hspace{0.25in} & C - B & = 1\\ {x^2} & : \hspace{0.25in} & 8A + 4B - C + D & = 10\\ {x^1} & : \hspace{0.25in} & - 4B + 4C - D + E & = 3\\ {x^0} & :\hspace{0.25in} & 16A - 4C - E & = 36\end{align*} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,A = 2,\,B = - 2,\,C = - 1,\,D = 1,\,E = 0\] <p>Don鈥檛 get excited if some of the coefficients end up being zero. It happens on occasion.</p> <p>Here鈥檚 the integral.</p> \[\begin{align*}\int{{\frac{{{x^3} + 10{x^2} + 3x + 36}}{{\left( {x - 1} \right){{\left( {{x^2} + 4} \right)}^2}}}\,dx}} & = \int{{\frac{2}{{x - 1}} + \frac{{ - 2x - 1}}{{{x^2} + 4}} + \frac{x}{{{{\left( {{x^2} + 4} \right)}^2}}}\,dx}}\\ & = \int{{\frac{2}{{x - 1}} - \frac{{2x}}{{{x^2} + 4}} - \frac{1}{{{x^2} + 4}} + \frac{x}{{{{\left( {{x^2} + 4} \right)}^2}}}\,dx}}\\ & = 2\ln \left| {x - 1} \right| - \ln \left| {{x^2} + 4} \right| - \frac{1}{2}{\tan ^{ - 1}}\left( {\frac{x}{2}} \right) - \frac{1}{2}\frac{1}{{{x^2} + 4}} + c\end{align*}\] </div> </div> </div> <p>To this point we鈥檝e only looked at rational expressions where the degree of the numerator was strictly less that the degree of the denominator. Of course, not all rational expressions will fit into this form and so we need to take a look at a couple of examples where this isn鈥檛 the case.</p> <a class="anchor" name="Int_PartFrac_Ex5"></a> <div class="example"> <span class="example-title">Example 5</span> Evaluate the following integral. \[\int{{\frac{{{x^4} - 5{x^3} + 6{x^2} - 18}}{{{x^3} - 3{x^2}}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln5" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln5" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln5" class="soln-content"> <p>So, in this case the degree of the numerator is 4 and the degree of the denominator is 3. Therefore, partial fractions can鈥檛 be done on this rational expression.</p> <p>To fix this up we鈥檒l need to do long division on this to get it into a form that we can deal with. Here is the work for that.</p> \[\require{enclose} \begin{align*} &\,\,\, x - 2\\ {x^3} - 3{x^2} & \enclose{longdiv}{{x^4} - 5{x^3} + 6{x^2} - 18} \\ - & \underline {\left( {{x^4} - 3{x^3}} \right)\hspace{0.9in}} \\ & \hspace{0.3in} - 2{x^3} + 6{x^2} - 18\\ & \underline { \hspace{0.08in}-\left( { - 2{x^3} + 6{x^2}} \right) \hspace{0.4in}} \\ & \hspace{1.35in} - 18\end{align*}\] <p>So, from the long division we see that,</p> \[\frac{{{x^4} - 5{x^3} + 6{x^2} - 18}}{{{x^3} - 3{x^2}}}\, = x - 2 - \frac{{18}}{{{x^3} - 3{x^2}}}\] <p>and the integral becomes,</p> \[\begin{align*}\int{{\frac{{{x^4} - 5{x^3} + 6{x^2} - 18}}{{{x^3} - 3{x^2}}}\,dx}} & = \int{{x - 2 - \frac{{18}}{{{x^3} - 3{x^2}}}\,dx}}\\ & = \int{{x - 2\,dx}} - \int{{\frac{{18}}{{{x^3} - 3{x^2}}}\,dx}}\end{align*}\] <p>The first integral we can do easily enough and the second integral is now in a form that allows us to do partial fractions. So, let鈥檚 get the general form of the partial fractions for the second integrand.</p> \[\frac{{18}}{{{x^2}\left( {x - 3} \right)}} = \frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{x - 3}}\] <p>Setting numerators equal gives us,</p> \[18 = Ax\left( {x - 3} \right) + B\left( {x - 3} \right) + C{x^2}\] <p>Now, there is a variation of the method we used in the first couple of examples that will work here. There are a couple of values of $x$ that will allow us to quickly get two of the three constants, but there is no value of $x$ that will just hand us the third.</p> <p>What we鈥檒l do in this example is pick $x$鈥檚 to get the two constants that we can easily get and then we鈥檒l just pick another value of $x$ that will be easy to work with (<em>i.e.</em> it won鈥檛 give large/messy numbers anywhere) and then we鈥檒l use the fact that we also know the other two constants to find the third.</p> \[\begin{align*}x & = 0 : & \hspace{0.25in} 18 & = B\left( { - 3} \right) & \hspace{0.15in}\Rightarrow \hspace{0.25in}B & = - 6\\ x & = 3 : & \hspace{0.25in} 18 & = C\left( 9 \right) & \hspace{0.15in} \Rightarrow \hspace{0.25in}C & = 2\\ x & = 1 : & 18 & = A\left( { - 2} \right) + B\left( { - 2} \right) + C = - 2A + 14 & \hspace{0.15in} \Rightarrow \hspace{0.25in}A & = - 2\end{align*}\] <p>The integral is then,</p> \[\begin{align*}\int{{\frac{{{x^4} - 5{x^3} + 6{x^2} - 18}}{{{x^3} - 3{x^2}}}\,dx}} & = \int{{x - 2\,dx}} - \int{{ - \frac{2}{x} - \frac{6}{{{x^2}}} + \frac{2}{{x - 3}}\,dx}}\\ & = \frac{1}{2}{x^2} - 2x + 2\ln \left| x \right| - \frac{6}{x} - 2\ln \left| {x - 3} \right| + c\end{align*}\] </div> </div> </div> <p>In the previous example there were actually two different ways of dealing with the ${x^2}$ in the denominator. One is to treat it as a quadratic which would give the following term in the decomposition</p> \[\frac{{Ax + B}}{{{x^2}}}\] <p>and the other is to treat it as a linear term in the following way,</p> \[{x^2} = {\left( {x - 0} \right)^2}\] <p>which gives the following two terms in the decomposition,</p> \[\frac{A}{x} + \frac{B}{{{x^2}}}\] <p>We used the second way of thinking about it in our example. Notice however that the two will give identical partial fraction decompositions. So, why talk about this? Simple. This will work for ${x^2}$, but what about ${x^3}$ or ${x^4}$? In these cases, we really will need to use the second way of thinking about these kinds of terms.</p> \[{x^3}\,\,\, \Rightarrow \,\,\,\,\,\frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{{x^3}}}\hspace{0.5in}{x^4}\,\,\, \Rightarrow \,\,\,\,\,\frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{{x^3}}} + \frac{D}{{{x^4}}}\] <p>Let鈥檚 take a look at one more example.</p> <a class="anchor" name="Int_PartFrac_Ex6"></a> <div class="example"> <span class="example-title">Example 6</span> Evaluate the following integral. \[\int{{\frac{{{x^2}}}{{{x^2} - 1}}\,dx}}\] <div class="example-content"> <span id="SHLink_Soln6" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln6" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln6" class="soln-content"> <p>In this case the numerator and denominator have the same degree. As with the last example we鈥檒l need to do long division to get this into the correct form. We鈥檒l leave the details of that to you to check.</p> \[\int{{\frac{{{x^2}}}{{{x^2} - 1}}\,dx}} = \int{{1 + \frac{1}{{{x^2} - 1}}\,dx}} = \int{{dx}} + \int{{\frac{1}{{{x^2} - 1}}\,dx}}\] <p>So, we鈥檒l need to partial fraction the second integral. Here鈥檚 the decomposition.</p> \[\frac{1}{{\left( {x - 1} \right)\left( {x + 1} \right)}} = \frac{A}{{x - 1}} + \frac{B}{{x + 1}}\] <p>Setting numerator equal gives,</p> \[1 = A\left( {x + 1} \right) + B\left( {x - 1} \right)\] <p>Picking value of $x$ gives us the following coefficients.</p> \[\begin{align*}x & = - 1 : & \hspace{0.25in} 1 & = B\left( { - 2} \right) & \hspace{0.25in} \Rightarrow \hspace{0.5in}B & = - \frac{1}{2}\\ x & = 1 \,\,\,\, : & \hspace{0.25in}1 & = A\left( 2 \right) & \hspace{0.25in} \Rightarrow \hspace{0.5in}A & = \frac{1}{2}\end{align*}\] <p>The integral is then,</p> \[\begin{align*}\int{{\frac{{{x^2}}}{{{x^2} - 1}}\,dx}} & = \int{{dx}} + \int{{\frac{{\frac{1}{2}}}{{x - 1}} - \frac{{\frac{1}{2}}}{{x + 1}}\,dx}}\\ & = x + \frac{1}{2}\ln \left| {x - 1} \right| - \frac{1}{2}\ln \left| {x + 1} \right| + c\end{align*}\] </div> </div> </div> </div>  <div class="footer"> <div class="footer-links"> [<a href="/Contact.aspx">Contact Me</a>] [<a href="/Privacy.aspx">Privacy Statement</a>] [<a href="/Help.aspx">Site Help & FAQ</a>] [<a href="/Terms.aspx">Terms of Use</a>] </div> <div class="footer-dates"> <div class="footer-copyright"><span id="lblCopyRight">© 2003 - 2024 Paul Dawkins</span></div> <div class="footer-spacer"></div> <div class="footer-modified-date">Page Last Modified : <span id="lblModified">11/16/2022</span></div> </div> </div> </div>  </body> </html>

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Calculus II - Partial Fractions