Evaluate the following integrals:

Given

Expressing the integral

Consider

By partial fraction decomposition,

⇒ 2x + 1 = Ax + B(x – 1)

⇒ 2x + 1 = Ax + Bx – B

⇒ 2x + 1 = (A + B)x – B

∴ B = -1 and A + B = 2

∴ A = 2 + 1 = 3

Thus,

Consider

Substitute u = x – 1 → dx = du.

We know that

Then,

∴

Then,

We know that

∴