Problem Statement
Evaluate the indefinite integral:
$\int \frac{2x}{(x^2 + 1)(x^2 + 3)} \, dx$
Verified Solution & Marking Scheme
Substitute u = x²
Let $u = x^2 \implies du = 2x \, dx$. The integral becomes $\int \frac{du}{(u + 1)(u + 3)}$.
Decompose into Partial Fractions
$\frac{1}{(u + 1)(u + 3)} = \frac{1}{2} \left( \frac{1}{u + 1} - \frac{1}{u + 3} \right)$
Integrate and Substitute Back
$\frac{1}{2} [\ln|u + 1| - \ln|u + 3|] + C = \frac{1}{2} \ln\left( \frac{x^2 + 1}{x^2 + 3} \right) + C$