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

The Correct Option is A

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 \( u \):
\[ \frac{\pi}{n} \log_e |x^n - 1| + C \]