Question

8. x tan-1 x

Answer 1

The integral is given as,

I= ∫ x tan −1 xdx

Use integration by parts. Consider tan −1 xas first function and xas second function.

I= ∫ x tan −1 xdx = tan −1 x ∫ xdx − ∫ ( ( d dx tan −1 x ) ∫ xdx )dx = x 2 2 tan −1 x− ∫ ( 1 1+ x 2 × x 2 2 )dx = x 2 2 tan −1 x− 1 2 ∫ 1+ x 2 −1 1+ x 2 dx

On further simplification, we get.

I= x 2 2 tan −1 x− 1 2 ( ∫ 1+ x 2 1+ x 2 dx− ∫ dx 1+ x 2 ) = x 2 2 tan −1 x− 1 2 ∫ dx + 1 2 tan −1 x = x 2 2 tan −1 x− x 2 + tan −1 x 2 +C I= 1 2 ( x 2 +1 ) tan −1 x− x 2 +C