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Evaluate: int(x^2tan^(-1)x)/(1+x^2)\ dx...

Evaluate: `int(x^2tan^(-1)x)/(1+x^2)\ dx`

Text Solution

Verified by Experts

Let `x=tan theta.`
` :. dx=sec^(2) theta d theta`
` :. I=int (x^(2)tan^(-1)x)/(1+x^(2))dx`
`=int (theta tan^(2) theta)/(1+tan^(2)theta)*sec^(2) theta d theta`
`= int theta tan^(2) theta d theta`
`=int theta (sec^(2) theta-1) d theta`
`=int theta sec^(2) theta d theta - int theta d theta`
`=theta tan theta -int 1*tan theta d theta-(theta^(2))/(2) +c`
`=theta tan theta -log_(e)|cos theta|-(theta^(2))/(2)+c, " where " theta = tan^(-1)x`.
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