∫1/(x(x^5 - 1)) dx is equal to

The correct answer is Option (1) → \(\displaystyle\frac{1}{5}\log_e\left|\frac{x^5 - 1}{x^5}\right| + C\): C is an arbitrary constant

Multiplying the numerator and the denominator by \(x^4\) makes the derivative of \(x^5\) appear:

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

Let \(t = x^5\), so \(dt = 5x^4\,dx\):

    \( \displaystyle I = \frac{1}{5}\int\frac{dt}{t(t - 1)} \)

By partial fractions:

    \( \displaystyle \frac{1}{t(t - 1)} = \frac{1}{t - 1} - \frac{1}{t} \)

So

    \( \displaystyle I = \frac{1}{5}\left(\log_e|t - 1| - \log_e|t|\right) + C \)

    \( \displaystyle = \frac{1}{5}\log_e\left|\frac{x^5 - 1}{x^5}\right| + C \)