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Evaluate int(sqrt(x^(2)+1){log(e)(x^(2)+...

Evaluate `int(sqrt(x^(2)+1){log_(e)(x^(2)+1)-2logx}dx)/(x^(4))`.

Text Solution

Verified by Experts

`I=int(sqrt(x^(2)+1)*log((x^(2)+1)/x^(2))dx)/(x^(4))`
`=int(sqrt(1+(1)/(x^(2)))*log(1+(1)/(x^(2)))dx)/(x^(3))`
Let `1+(1)/(x^(2))=t`
` :. -(2)/(x^(3))dx=dt`
` :. I=-(1)/(2)int sqrt(t)log t dt`
`= -(1)/(2)[log t*(t^(3//2))/(3//2)-int (1)/(t)*(t^(3//2))/(3//2)dt]`
`= -(1)/(2)[(2)/(3)t^(3//2)log_(e)t-(2)/(3)(t^(3//2))/(3//2)]+c`
`=-(1)/(3)t^(3//2) log_(e)t+(2)/(9)t^(3//2)+c, " where " t=1+(1)/(x^(2))`
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