Consider,
I = \(\int\frac{x^2+x-1}{x^2+x-6}\)dx
Expressing the integral,
Factorizing the denominator,
⇒ 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,
Then,