Evaluate: `intx/((a^2-x^2)^(3setminus2))\ dx` .

Explanation

Let `I=int frac{x^{2}}{(a^{2}-x^{2})^{3 / 2}} d x`

Let `x=a cos theta`

On differentiating both sides, we get


`d x=-a sin theta d theta `

`therefore I=int frac{a^{2} cos ^{2} theta}{(a^{2}-a^{2} cos ^{2} theta)^{frac{3}{2}}} times-a sin theta d theta `

`=-int frac{a^{3} cos ^{2} theta sin theta}{a^{3}(1-cos ^{2} theta)^{frac{3}{2}} d theta} `

`=-int frac{cos ^{2} theta sin theta}{sin ^{3} theta} d theta `

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