$\int \frac{\pi}{x^{n+1}-x} d x=$

$\frac{\pi}{n} \log _e\left|\frac{x^n-1}{x^n}\right|+C$

The correct answer is Option (1) → $\frac{\pi}{n} \log _e\left|\frac{x^n-1}{x^n}\right|+C$

$\int \frac{\pi}{x^{n+1}-x} d x$

$=\int\frac{πx^{-(n+1)}}{1-x^{-n}}dx$

so $y=1-x^{-n}$

$dy=nx^{-(n+1)}dx$

so $I=\int\frac{π}{n}\frac{1}{y}dy$

$=\frac{π}{n}\log y+c$

$=\frac{π}{n}\log(1-x^{-n})+c$

$=\frac{π}{n}\log\left|\frac{x^n-1}{x^n}\right|+c$