Question

Find \(\int \frac{dx}{(x+1)(x+2)}\).

(a) \(Log \left|\frac{x+1}{x+2}\right|+ C\)

(b) \(Log \left|\frac{x-1}{x+2}\right|+ C\)

(c) \(Log \left|\frac{x+2}{x+1}\right|+ C\)

(d) \(Log \left|\frac{x+1}{x-2}\right|+ C\)

This question was addressed to me during an online interview.

Asked question is from Integration by Partial Fractions topic in section Integrals of Mathematics – Class 12

Answer 1

The correct answer is (a) \(Log \left|\frac{x+1}{x+2}\right|+ C\)

The best I can explain: It is a proper rational function. Therefore,

\(\frac{1}{(x+1)(x+2)} = \frac{A}{(x+1)} + \frac{B}{(x+2)}\)

Where real numbers are determined, 1 = A(x+2) + B(x+1), Equating coefficients of x and the constant term, we get A+B = 0 and 2A+B = 1. Solving it we get A=1, and B=-1.

Thus, it simplifies to, \(\frac{1}{(x+1)} + \frac{-1}{(x+2)} = \int \frac{dx}{(x+1)} – \int \frac{dx}{(x+2)}\).

= log|x+1| – log|x+2| + C

= \(Log \left|\frac{x+1}{x+2}\right|+ C\).