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