Question

Evaluate the following integrals:

∫ x cos^-1 x dx

Answer 1

Let x = cos t ; t = cos^{- 1}x

dx = - sin t dt

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking first function to the one which comes first in the list.

Here t is first function and sin 2t as second function.

We know that cos 2t = 2 cos²t - 1 and sin 2t = 2 sin t × cos t and sin t = √1 - cos 2t

Replacing in above equation