Problem Statement
Evaluate the indefinite integral:
$\int \frac{dx}{x(x^5 + 1)}$
Verified Solution & Marking Scheme
Multiply Numerator and Denominator by x⁴
To prepare for substitution, multiply numerator and denominator by $x^4$:
$\int \frac{dx}{x(x^5 + 1)} = \int \frac{x^4}{x^5 (x^5 + 1)} \, dx$
Substitute t = x⁵
Let $t = x^5 \implies dt = 5x^4 \, dx \implies x^4 \, dx = \frac{1}{5} dt$.
$\int \frac{x^4}{x^5(x^5 + 1)} \, dx = \frac{1}{5} \int \frac{dt}{t(t + 1)}$
Decompose into Partial Fractions and Integrate
Using the algebraic identity $\frac{1}{t(t + 1)} = \frac{1}{t} - \frac{1}{t + 1}$:
$\frac{1}{5} \int \left( \frac{1}{t} - \frac{1}{t + 1} \right) \, dt = \frac{1}{5} [\ln|t| - \ln|t + 1|] + C$
$= \frac{1}{5} \ln \left| \frac{t}{t + 1} \right| + C$
Substituting back $t = x^5$:
$= \frac{1}{5} \ln \left| \frac{x^5}{x^5 + 1} \right| + C$.