Evaluate the following integrals:

We can write it as

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 cot ^{- 1}x is first function and 1 as the second function.

Let 1 + x² = t

2xdx = dt

Xdx = dt/2

Now replacing t with 1 + x²

= xcot ^{- 1}x + log(1 + x²)/2 + c