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Question

Evaluate the following integrals
x2(a2x2)3/2dx

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Solution

put x=asinθ
sinθ=xa
dx=acosθdθ
cosθ1x2a2=a2x2a
I=a2sin2θ×acosθdθ(a2(1sin2θ))3/2
a3sin2θcosθdθa3cos3θ
Itan2θdθ
I=tan1θdθ
I=1cos2θcos2θdθ=sec2θdθdθ
=tanθθ+C
I=xa2x2sin1(xa)+C

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