Evaluate: `int(x^7)/((a-x^2)^5)\ dx` .

Explanation

Suppose `x=a sin theta`

By differentiating both sides


`d x=a cos theta d theta `

`therefore I=int frac{a^{8} sin ^{7} theta cos theta}{(a^{2}-a^{2} sin ^{2} theta)^{5}} d theta `

`=int frac{a^{8} sin ^{7} theta cos theta}{a^{10}(1-sin ^{2} theta)^{5}} d theta `

`=int frac{sin ^{7} theta}{a^{2} cos ^{9} theta} d theta `

`=frac{1}{a^{2}} int tan ^{7} theta sec ^{2} theta d theta`

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