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

dx = - sin 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 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 cos2t = 2cos²t - 1 and sin2t = 2sint×cost and sint = √1 - cos²t

Replacing in above equation