`int_(0)^(1)((tan^(-1)x)^(2))/(1+x^(2))dx` का मान ज्ञात कीजिए ।

Explanation

माना `tan^(-1)x=t`
`rArr" "(1)/(1+x^(2))dx=dt`
`x=0" पर "t=tan^(-1)0=0`
`x=1" पर "t=tan^(-1)1=pi//4`
`therefore" "int_(0)^(1)(tan^(-1)x)/(1+x^(2))dx`
`" "=int_(90)^(pi//4)t^(2)dt=[(1)/(3)t^(3)]_(0)^(pi//4)`
`" "=(1)/(3)((pi)/(4))^(3)-0=(pi^(3))/(192)`