Question:

<p><strong>Step 1:</strong> Set up partial fractions decomposition:</p><p>$\frac{1}{(x + 2)(x^2 + 1)} = \frac{A}{x + 2} + \frac{Bx + C}{x^2 + 1}$</p><p><strong>Step 2:</strong> Multiply both sides by $(x + 2)(x^2 + 1)$:</p><p>$1 = A(x^2 + 1) + (Bx + C)(x + 2)$</p><p><strong>Step 3:</strong> Substitute $x = -2$:</p><p>$1 = A(4 + 1) = 5A$</p><p>$A = \frac{1}{5}$</p><p><strong>Step 4:</strong> Compare coefficients of $x^2$ and constant term:</p><p>Coefficient of $x^2$: $0 = A + B \Rightarrow B = -\frac{1}{5}$</p><p>Constant term: $1 = A + 2C \Rightarrow 1 = \frac{1}{5} + 2C \Rightarrow C = \frac{2}{5}$</p><p><strong>Step 5:</strong> Integrate:</p><p>$\int \frac{1}{(x + 2)(x^2 + 1)}dx = \frac{1}{5}\int\frac{dx}{x+2} - \frac{1}{5}\int\frac{x dx}{x^2+1} + \frac{2}{5}\int\frac{dx}{x^2+1}$</p><p>$= \frac{1}{5}\log|x+2| - \frac{1}{10}\log(x^2+1) + \frac{2}{5}\tan^{-1}x + C$</p><p>∴ Answer is (a).</p>