Problem Statement
Evaluate the indefinite integral:
$\int \frac{1}{x(x^4 + 1)} \, dx$
Verified Solution & Marking Scheme
Multiply Numerator and Denominator by x³
$\int \frac{x^3}{x^4(x^4 + 1)} \, dx$
Substitute u = x⁴
Let $u = x^4 \implies du = 4x^3 dx \implies x^3 dx = \frac{du}{4}$:
$= \frac{1}{4} \int \frac{du}{u(u + 1)}$
Decompose into Partial Fractions and Integrate
$\frac{1}{u(u + 1)} = \frac{1}{u} - \frac{1}{u + 1}$
$\frac{1}{4} \int \left(\frac{1}{u} - \frac{1}{u + 1}\right) du = \frac{1}{4} [\ln|u| - \ln|u + 1|] + C = \frac{1}{4} \ln\left|\frac{u}{u + 1}\right| + C$
Substitute back $u = x^4$:
$= \frac{1}{4} \ln\left(\frac{x^4}{x^4 + 1}\right) + C$