Question

7. x sim 'x

Answer 1

The integral is given below as,

I= ∫ x sin −1 xdx

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

I= ∫ x sin −1 xdx I= sin −1 x ∫ xdx − ∫ ( d dx sin −1 x ∫ xdx ) dx

On integrating, we get

I= x 2 2 sin −1 x+ 1 2 ∫ − x 2 1− x 2 dx = x 2 2 sin −1 x+ 1 2 ∫ ( 1− x 2 1− x 2 − 1 1− x 2 )dx = x 2 2 sin −1 x+ 1 2 [ ∫ 1− x 2 dx − ∫ dx 1− x 2 ] I= x 2 2 sin −1 x+ 1 2 [ x 2 1− x 2 + 1 2 sin −1 x− sin −1 x ]+C

By simplifying further, we get

I= x 2 2 sin −1 x+ 1 2 [ x 2 1− x 2 − 1 2 sin −1 x ]+C = x 2 2 sin −1 x− 1 4 sin −1 x+ x 4 1− x 2 +C I= 1 4 ( 2 x 2 −1 ) sin −1 x+ x 4 1− x 2 +C