Question

9. x cos-1 x

Answer 1

Let the integral be,

I= ∫ x cos −1 xdx

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

I= ∫ x cos −1 xdx I= cos −1 x ∫ xdx − ∫ ( d dx cos −1 x ∫ xdx )dx I= x 2 cos −1 x 2 − 1 2 ∫ { −1 1− x 2 × x 2 2 }dx I= x 2 cos −1 x 2 − 1 2 ∫ 1− x 2 −1 1− x 2 dx

Simplify further,

I= x 2 cos −1 x 2 − 1 2 [ ∫ 1− x 2 1− x 2 dx− ∫ dx 1− x 2 ] = x 2 cos −1 x 2 − 1 2 ∫ 1− x 2 dx− 1 2 ∫ −dx 1− x 2 = x 2 cos −1 x 2 − 1 2 I 1 − 1 2 cos −1 x+C

Consider I 1 = ∫ 1− x 2 dx.

Again use integration by parts,

I 1 =x 1− x 2 − ∫ d dx 1− x 2 ∫ xdx =x 1− x 2 − ∫ −2x 2 1− x 2 xdx =x 1− x 2 − ∫ − x 2 1− x 2 xdx =x 1− x 2 − ∫ 1− x 2 −1 1− x 2 dx

Further simplify,

I 1 =x 1− x 2 −{ ∫ 1− x 2 dx+ ∫ −dx 1− x 2 } =x 1− x 2 −{ I 1 + cos −1 x } 2 I 1 =x 1− x 2 − cos −1 x I 1 = x 2 1− x 2 − 1 2 cos −1 x

Substitute the values in I.

I= x 2 cos −1 x 2 − x 4 1− x 2 + 1 4 cos −1 x− 1 2 cos −1 x+C = x 2 cos −1 x 2 − 1 4 cos −1 x− x 4 1− x 2 +C I= 1 4 ( 2 x 2 −1 ) cos −1 x− x 4 1− x 2 +C

Thus, the integration of ∫ x cos −1 xdx is 1 4 ( 2 x 2 −1 ) cos −1 x− x 4 1− x 2 +C.