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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»

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

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


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