Question

Evaluate the following integrals:

∫ x sin^-1 x dx

Answer 1

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

dx = cos t dt

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

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

We know that cos 2t = 1 - 2 sin²t , sin2t = 2 sin t × cos t and cos t = √1 - sin²t

Replacing in above equation