Evaluate the following integrals:

Let sin ^{- 1}x = t

X = sint

Putting this in the original equation, we get

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 t as the second function.

We can write cos t = √1 - sin²t

= - t(√1 - sin²t) + sint + c

Now replacing sin ^{- 1}x = t

= - sin ^{- 1}x(√1 - x²) + x + c