∫(3x+2)/((x+1)(x+2)) dx is equal to

The correct answer is Option (1) → \(-\log|x+1|+4\log|x+2|+C\)

By partial fractions, let

    \( \displaystyle \frac{3x+2}{(x+1)(x+2)} = \frac{A}{x+1}+\frac{B}{x+2} \)

    \( 3x+2 = A(x+2)+B(x+1) \)

Putting x = −1:

    \( -1 = A \)

Putting x = −2:

    \( -4 = -B \), so \( B = 4 \)

So

    \( \displaystyle \int\frac{3x+2}{(x+1)(x+2)}\,dx = \int\left(\frac{-1}{x+1}+\frac{4}{x+2}\right)dx \)

    \( = -\log|x+1|+4\log|x+2|+C \)