CA Foundation · Quantitative Aptitude · Differential and Integral Calculus
Evaluate ∫ (x + 5)/((x + 1)(x + 2)) dx (C is the constant of integration).
The integral is 4 log|x+1| − 3 log|x+2| + C. Partial fractions give A = 4 at x = −1 and B = −3 at x = −2, so the integrand splits into 4/(x+1) − 3/(x+2), each integrating to a logarithm.
- A4 log|x + 1| − 3 log|x + 2| + CCorrect
- B3 log|x + 1| − 4 log|x + 2| + C
- C4 log|x + 1| + 3 log|x + 2| + C
- Dlog|(x + 1)(x + 2)| + C
Explanation
Write (x+5)/((x+1)(x+2)) = A/(x+1) + B/(x+2). Putting x = −1 gives A = 4/1 = 4. Putting x = −2 gives B = 3/(−1) = −3. Hence the integral is 4 log|x+1| − 3 log|x+2| + C. Swapping the coefficients, as in the second option, is wrong because A and B belong to the other factors.
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