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in Indefinite Integral by (29.4k points)
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Evaluate the following integrals :

\(\int\frac{x^2+x-1}{x^2+x-6}\)dx

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Best answer

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,

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