Evaluate the following integrals:

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

dx = cos t dt

We know that sin 2t = 2 sint×cost

We have

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 cos2t = 1 - 2sin²t , sin2t = 2sint×cost and cos t = √1 - sin²t

Replacing in above equation