Evaluate the following integrals:

Given

Rewriting, we get

Expressing the integral

Consider

By partial fraction decomposition,

⇒ x – 2 = A (2x – 1) + Bx

⇒ x – 2 = 2Ax – A + Bx

⇒ x – 2 = (2A + B) x – A

∴ A = 2 and 2A + B = 1

∴ B = 1 – 4 = -3

Thus,

Consider

We know that

And consider

Let u = 2x – 1 → dx = 1/2 du

We know that

Then,

Then,

We know that