Question

Evaluate the integral $\int \frac{\pi}{x^n + 1 - x} , dx$:

• A. $\frac{\pi}{n} \log_e \left| \frac{x^n - 1}{x^n} \right| + C$
• B. $\log_e \left| \frac{x^{n+1} + 1}{x^{n-1}} \right| + C$
• C. $\frac{\pi}{n} \log_e \left| \frac{x^{n+1}}{x^n} \right| + C$
• D. $\pi \log_e \left| \frac{x^n}{x^{n-1}} \right| + C$

Answer 1

Answer: (A)

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

Answer 2

Begin with the integral:
$\int \frac{\pi}{x^{n+1} - x} \, dx$
Factor the denominator:
$x^{n+1} - x = x \cdot (x^n - 1)$
Thus, the integral becomes:
$\int \frac{\pi}{x \cdot (x^n - 1)} \, dx$
Use the substitution$u = x^n - 1$, so $du = n x^{n-1} \, dx$, which gives $dx = \frac{du}{n x^{n-1}}$.
Substitute $u$ and simplify:
$\int \frac{\pi}{x \cdot u} \cdot \frac{du}{n x^{n-1}} = \frac{\pi}{n} \int \frac{1}{u} \cdot \frac{1}{x^n} \, du$
Since $x^n = u + 1$, we get:
$\frac{\pi}{n} \int \frac{1}{u} \, du = \frac{\pi}{n} \log_e |u| + C$
Substitute back for $u$:
$\frac{\pi}{n} \log_e |x^n - 1| + C$