CK12 Calculus - Single Variable
This is the formulafor integrationby parts.With the properchoiceof and the secondintegralmay be easierto integrate.The following ...
Then Substitutingbackinto the formulato integrate,we get As you can see, this integralis worsethan whatwe startedwith! This tell ...
We take the differentialof and the simplestantiderivativeof : Substitutingbackinto the formula, We havemadethe right choicebecau ...
RepeatedUse of Integrationby Parts Oftentimeswe use integrationby partsmorethan onceto evaluatethe integral,as the examplebelows ...
Solution: Beginas usualby letting and Next,createa tablethat consistsof threecolumns, as shownbelow: Alternatesigns tivesand its ...
Noticethat the unknownintegralnow appearson both sidesof the equation.We can simplymovethe unknown integralon the right to the l ...
6. 7. Use both the methodof -substitutionand the methodof integrationby partsto integratethe integral below. Both methodswill ...
9. 10. 11. Integrationby PartialFractions LearningObjectives A studentwill be able to: Computeby handthe integralsof a wide var ...
Solution: We beginby factoringthe denominatoras Thenwritethe partialfraction decompositionas Our goal at this pointis to find th ...
To find the partialfractiondecompositionof a properrationalfunction, factorthe denominator and writean equationthat has the for ...
The integralwill become wherewe haveused -substitutionfor the secondintegral. Example3: Evaluate. Solution: We beginby factoring ...
To find we can simplysubstituteany valueof alongwith the valuesof and obtained. Choose : Now we havesolvedfor and We use the par ...
For Thusour integralbecomes Integratingand substitutingthe limits, ReviewQuestions Evaluatethe followingintegrals. ...
Evaluatethe integralby makingthe proper -substitutionto convertto a rationalfunction: Find the area underthe curve over the int ...
TrigonometricIntegrals LearningObjectives A studentwill be able to: Computeby handthe integralsof a wide varietyof functionsby ...
Example2: Evaluate Solution: Integratingterm by term, Example3: Evaluate Solution: ...
Recallthat so by substitution, The first integralshouldbe straightforward.The secondcan be doneby the methodof -substitutionby l ...
Referringto the tableagain,we can now substitute in the integral: Example5: Evaluate. Solution: Here, We followthe third procedu ...
IntegratingPowersof Secantsand Tangents In this sectionwe will studymethodsof integratingfunctionsof the form where and are nonn ...
Use -substitution.Let then the integralbecomes, Thereare two reductionformulasthat help evaluatehigherpowersof tangentand secant ...
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