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