Integrals

Board: JEE Main |Exam:JEEMain

JEEMainstudynotesforIntegralsbyAjayYadav.

KeyConcepts

IndefiniteIntegrals

Standardforms.Substitution.Integrationbyparts.Partialfractions.Specialintegrals:∫1/(x²+a²)dx,∫1/√(a²-x²)dx,∫√(a²-x²)dx.

DefiniteIntegrals

Fundamentaltheorem.Properties:∫₀^af(x)dx=∫₀^af(a-x)dx.∫₀²f(x)dx=2∫₀^af(x)dxiff(2a-x)=f(x).∫₀²f(x)dx=0iff(2a-x)=-f(x).

AreaUnderCurves

A=∫|f(x)-g(x)|dx.Areabetweentwocurves.

ReductionFormulas

∫sinⁿxdx,∫cosⁿxdx.Wallisformula.

ImportantFormulas

∫1/(x²+a²)dx	(1/a)tan⁻¹(x/a)+C
∫1/√(a²-x²)dx	sin⁻¹(x/a)+C
Evenfunction	∫₀²f(x)dx=2∫₀af(x)dx
Oddfunction	∫₀²f(x)dx=0

SolvedExamples

Example1:Evaluate∫1/(4+x²)dx.

Solution:(1/2)tan⁻¹(x/2)+C.

Example2:Evaluate∫₀⁼/₂cosxdx.

Solution:[sinx]₀⁼/₂=1-0=1.

PracticeQuestions

1. ∫xcosxdx.
2. ∫1/(x²-4)dx.
3. ∫₀¹x²e^xdx.
4. ∫₀⁼|cosx|dx.
5. ∫1/(x²+2x+5)dx.

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