If ∫ - 1 3 1 / 3 x 4 1 - x 4 cos -1 2 x 1 + x 2 dx = k ∫ 0 1 / 3 x 4 1 - x 4 dx , then the value of k is equal to

Step-by-step solution

Let I = ∫ - 1 3 1 / 3 x 4 1 - x 4 cos -1 2 x 1 + x 2 dx = ∫ - 1 3 1 / 3 x 4 1 - x 4 π 2 - sin -1 2 x 1 + x 2 dx = π 2 ∫ − 1 / 3 1 / 3 x 4 1 − x 4 d x − ∫ − 1 / 3 1. 3 ( x 4 1 − x 4 ) sin − 1 ( 2 x 1 + x 2 ) d x ⏟ =0, as integrand is an odd function = π 2 . 2 ∫ 0 1 / 3 x 4 1 − x 4 d x ( a s x 4 1 − x 4 is an even function ) Hence, k = π