Evaluate the following integrals:

Consider

Expressing the integral

Consider

Let and split,

Consider

Let

We know that

Now consider

By partial fraction decomposition,

⇒ 1 = A (x – 2) + B (x – 3)

⇒ 1 = Ax – 2A + Bx – 3B

⇒ 1 = (A + B) x – (2A + 3B)

⇒ A + B = 0 and 2A + 3B = -1

Solving the two equations,

⇒ 2A + 2B = 0

2A + 3B = -1

-B = 1

∴ B = -1 and A = 1

Consider

Let u = x – 3 → dx = du

We know that

Similarly

Let u = x – 2 → dx = du

We know that

Then,

Then,

Then,

We know that