`(sqrt(x^(2)+1)[log(x^(2)+1)-2logx])/(x^(4))`

Explanation

माना
`int(sqrt(x^(2)+1)[log(x^(2)+1)-2logx])/(x^(4))dx`
`=intsqrt((x^(2)+1)/(x^(4)))[log(x^(2)+1)-logx^(2)]dx`
`=int(sqrt(x^(2)+1))/(x^(4))[log(1+(1)/(x^(2)))]dx`
`=int(1)/(x^(3))sqrt((x^(2)+1)/(x^(2)))[log(1+(1)/(x^(2)))]dx`
माना `" "(x^(2)+1)/(x^(2))=t`
`rArr" "1+(1)/(x^(2))=t" "rArr" "-(2)/(x^(3))dx`
`rArr" "(dx)/(x^(3))=-(1)/(2)dt`
`therefore" "I=intsqrttlogt.(-(dt)/(2))=-(1)/(2) int(t)^(1//2)logt dt`
`=-(1)/(2)[logt. intt^(1//2)dt-int{((d)/(dt)logt) int t^(1//2)dt}dt]`
`=-(1)/(2)[log t.(t^(3//2))/(3//2)-int(1)/(t).(t^(3//2))/((3)/(2))dt]`
`=-(1)/(2)[(2)/(3)log t. t^(3//2)-(2)/(3)int t^(1//2)dt]`
`=-(1)/(2)xx(2)/(3)[log t. t^(3//2)-intt^(1//2)dt]`
`=-(1)/(3)(log t.t^(3//2)-(t^(3//2))/(3//2))+C`
`=-(1)/(3)(log t.t^(3//2)-(t^(3//2))/(3//2))+C`
`=-(1)/(3)t^(3//2)[logt-(2)/(3)]+C`
`=-(1)/(3)(1+(1)/(x^(2)))^(3//2)[log((x^(2)+1)/(x^(2)))-(2)/(3)]+C`