Question

Evaluate : `int tan^-1x dx`

Answer 1

Let I = `int tan^-1 x dx`       

        = `int tan^-1x . 1dx`

Integrating by parts, we get

I = `tan^-1 x int 1 dx - int [d/dx ( tan^-1 x) . int 1.dx] dx`

= `tan^-1 x . x - int 1/[ 1 + x^2 ]. x dx`

= `x tan^-1 x - 1/2 int 2x/[ 1 + x^2 ] dx`

= `x tan^-1 x - 1/2 log |( 1 + x^2)| + c`

                           `( ∵ int f'(x)/f(x) dx = log | f(x) | + c)`