Evaluate the integral: $\int\limits_{1}^{2} \frac{x \, dx}{(x + 1)(x + 2)}$

$\ln\left( \frac{32}{27} \right)$

The correct answer is Option (1) → $\ln\left( \frac{32}{27} \right)$

Let $I = \int\limits_{1}^{2} \frac{x \, dx}{(x + 1)(x + 2)}$

Using partial fraction, we get $\frac{x}{(x + 1)(x + 2)} = \frac{-1}{x + 1} + \frac{2}{x + 2}$

So $\int \frac{x \, dx}{(x + 1)(x + 2)} = -\log |x + 1| + 2\log |x + 2| = F(x)$

Therefore, by the second fundamental theorem of calculus, we have

$I = F(2) - F(1) = [-\log 3 + 2 \log 4] - [-\log 2 + 2 \log 3]$

$= -3 \log 3 + \log 2 + 2 \log 4 = \log \left( \frac{32}{27} \right)$