Evaluate ∫ (1)/(x(x⁵ + 1)) , dx — JEE Mathematics

Multiply the numerator and denominator by $x^4$:

$$I = \int \frac{x^4}{x^5(x^5 + 1)} \, dx$$

Let $t = x^5$, then $dt = 5x^4 \, dx \implies x^4 \, dx = \frac{dt}{5}$.
Substitute into the integral:

$$I = \frac{1}{5} \int \frac{1}{t(t + 1)} \, dt$$

Using partial fractions: $\frac{1}{t(t+1)} = \frac{1}{t} - \frac{1}{t+1}$.

$$I = \frac{1}{5} \int \left(\frac{1}{t} - \frac{1}{t+1}\right) \, dt = \frac{1}{5} (\ln|t| - \ln|t+1|) + C = \frac{1}{5} \ln\left|\frac{t}{t+1}\right| + C$$

Substitute $t = x^5$ back:

$$I = \frac{1}{5} \ln\left|\frac{x^5}{x^5 + 1}\right| + C$$

Answer: $\frac{1}{5} \ln\left|\frac{x^5}{x^5 + 1}\right| + C$