Evaluate the following integrals:

Given

Expressing the integral

Consider

Let and split,

Consider

Let

We know that

Now consider

By partial fraction decomposition,

⇒ 1 = A (x + 2) + B (x + 5)

⇒ 1 = Ax + 2A + Bx + 5B

⇒ 1 = (A + B) x + (2A + 5B)

⇒ A + B = 0 and 2A + 5B = 1

Solving the two equations,

⇒ 2A + 2B = 0

2A + 5B = 1

-3B = -1

∴ B = 1/3 and A = -1/3

Consider

Let u = x + 2 → dx = du

We know that

Similarly

Let u = x + 5 → dx = du

We know that

Then,

Then,

Then,

We know that