Evaluate the following integrals:

[CBSE 2006C]

Tip – If f₁(x) and f₂(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as

where f₁(x) and f₂(x) are the first and second functions respectively.

Taking f₁(x) = cot^-1x and f₂(x) = x²,

Taking (1+x²)=a,

2xdx=da i.e. xdx=da/2

Again, x²=a-1

Replacing the value of a, we get,

The total integration yields as

, where c is the integrating constant