Let \(\displaystyle\frac{2x+5}{(x+1)(x+4)} = \frac{A}{x+1}+\frac{B}{x+4}\). Consider the following statements, where C is a constant of integration:
(A) A = 1
(B) B = −1
(C) \(\int\displaystyle\frac{2x+5}{(x+1)(x+4)}\,dx = \log\left|x^2+5x+4\right|+C\)
(D) \(\int\displaystyle\frac{2x+5}{(x+1)(x+4)}\,dx = \log\left|\frac{x+1}{x+4}\right|+C\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
Clearing the denominators in the partial fractions:
\( 2x+5 = A(x+4)+B(x+1) \)
Putting x = −1:
\( 3 = 3A,\quad A = 1 \)
Putting x = −4:
\( -3 = -3B,\quad B = 1 \)
So
\( \displaystyle \int\frac{2x+5}{(x+1)(x+4)}\,dx = \log|x+1|+\log|x+4|+C \)
\( = \log\left|x^2+5x+4\right|+C \)
This also follows directly, since \(2x+5\) is the derivative of the denominator \(x^2+5x+4\).
So A = 1, B = 1 and the integral is \(\log\left|x^2+5x+4\right|+C\).
Checking the statements
(A) A = 1 — Correct. At x = −1, 3 = 3A.
(B) B = −1 — Incorrect. At x = −4, −3 = −3B, so B = 1.
(C) \(\int\frac{2x+5}{(x+1)(x+4)}\,dx = \log\left|x^2+5x+4\right|+C\) — Correct. It is \(\log|x+1|+\log|x+4|\).
(D) \(\int\frac{2x+5}{(x+1)(x+4)}\,dx = \log\left|\frac{x+1}{x+4}\right|+C\) — Incorrect. This would need B = −1; with B = 1 the two logs are added.