Evaluate the following integrals:

Given

Rationalizing the denominator,

Integral is of form

Writing numerator as

⇒ px + q = λ(2ax + b) + μ

⇒ -x + 1 = λ (-2x) + μ

∴ λ = 1/2 and μ = 1

Let -x + 1 = 1/2(-2x) + 1 and split,

Consider

Let

We know that

Consider

We know that

Then,