`int(sqrt(x^(2)+1)[log(x^(2)+1)-2logx])/(x^(4))`dx is equal to

Explanation

We have
`I=int(sqrt(x^(2)+1))/(x^(4))*log((x^(2)+1)/(x^(2)))dx`
`=intsqrt(1+(1)/(x^(2)))*(1)/(x^(3))log(1+(1)/(x^(2)))dx` . . .(i)
Putting `(1+(1)/(x^(2)))=t and(-2)/(x^(3))dx=dt,i.e.,(1)/(x^(3))dx=-(1)/(2)dt` in (i), we get
`I=-(1)/(2)intunderset(I)((logt))underset(II)(sqrt(t))dt`
`=-(1)/(2){(2)/(3)(logt)t^(3//2)-(2)/(3)int((1)/(t)xxt^(3//2))dt}`
`=-(1)/(3)t^(3//2)(logt)+(1)/(3)intt^(1//2)dt`
`=-(1)/(3)t^(3//2)(logt)+(2)/(9)t^(3//2)+C`
`-(1)/(3)t^(3//2){logt-(2)/(3)}+C`
`=-(1)/(3)(1+(1)/(x^(2)))^(3//2){log(1+(1)/(x^(2)))-(2)/(3)}+C`.