If \(\int\displaystyle\frac{2x}{x^2+3x+2}\,dx=m\log|x+2|+n\log|x+1|+C\), then the values of m and n are:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(m=4, n=-2\)
Since \(x^2+3x+2 = (x+1)(x+2)\), use partial fractions:
\( \displaystyle \frac{2x}{(x+1)(x+2)} = \frac{A}{x+1}+\frac{B}{x+2} \)
\( 2x = A(x+2)+B(x+1) \)
Putting \(x = -1\):
\( -2 = A \)
Putting \(x = -2\):
\( -4 = -B \), so \( B = 4 \)
Integrating:
\( \displaystyle \int\frac{2x}{x^2+3x+2}\,dx = -2\log|x+1|+4\log|x+2|+C \)
Comparing with \(m\log|x+2|+n\log|x+1|+C\): \(m = 4\), \(n = -2\).
Why the other options are incorrect
Option (2) \(m=-2, n=4\) — Incorrect. The values are swapped: \(-2\) goes with \(\log|x+1|\) and 4 with \(\log|x+2|\).
Option (3) \(m=2, n=2\) — Incorrect. Differentiating gives \(\frac{2}{x+2}+\frac{2}{x+1} = \frac{4x+6}{x^2+3x+2}\), not \(\frac{2x}{x^2+3x+2}\).
Option (4) \(m=6, n=4\) — Incorrect. Differentiating gives \(\frac{6}{x+2}+\frac{4}{x+1} = \frac{10x+14}{x^2+3x+2}\), not \(\frac{2x}{x^2+3x+2}\).