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,
Substitute u = x – 1 → dx = du.
We know that
Then,
∴