Evaluate the following integrals:
Consider
Expressing the integral
Let x2 + x – 1 = x2 + x – 6 + 5
Consider
Factorizing the denominator,
By partial fraction decomposition,
⇒ 1 = A(x + 3) + B(x – 2)
⇒ 1 = Ax + 3A + Bx – 2B
⇒ 1 = (A + B) x + (3A – 2B)
⇒ Then A + B = 0 … (1)
And 3A – 2B = 1 … (2)
Solving (1) and (2),
2 × (1) → 2A + 2B = 0
1 × (2) → 3A – 2B = 1
5A = 1
∴ A = 1/5
Substituting A value in (1),
⇒ A + B = 0
⇒ 1/5 + B = 0
∴ B = -1/5
Thus,
Let x – 2 = u → dx = du
And x + 3 = v → dx = dv.
We know that
Then,
We know that
∴