`int_0^1(tan^(-1)x)/x dx` is equals to (a) `int_0^(pi/2)(sinx)/x dx` (b) `int_0^(pi/2)x/(sinx)dx` (c)`1/2int_0^(pi/2)(sinx)/x dx` (d) `1/2int_0^(pi/2)(""x)/(sinx)dx`

Explanation

`I=int_(0)^(1)(tan^(-1)x)/x dx`
Putting `x=tan theta`
`:.dx=sec^(2)theta d theta`
`:.I=int_(0)^(pi//4) (theta)/(tan theta).sec^(2) theta d theta`
`=int_(0)^(pi//4)(2theta)/(sin 2theta)d theta`
Putting `2theta=t, ,.2d theta=dt`,
`:. I=int_(0)^(pi//2)t/(sint) . (dt)/2`
`=1/2int_(0)^(pi//2)x/(sinx)dx`