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

Hint:

When dealing with integrals involving polynomial expressions, substitution can often simplify the process. If you recognize a part of the integrand that can be substituted (like \( x^n - 1 \) in this case), make the substitution and simplify the expression before performing the integration. Also, remember that after substitution, you might need to revert to the original variable to express the final answer.

Answer 1

The Correct Option is A

Start with the integral:
\[ \int \frac{\pi}{x^{n+1} - x} \, dx \]
Factor the denominator:
\[ x^{n+1} - x = x \cdot (x^n - 1) \]
The integral is now:
\[ \int \frac{\pi}{x \cdot (x^n - 1)} \, dx \]
Apply the substitution \( u = x^n - 1 \). This implies \( du = n x^{n-1} \, dx \), so \( dx = \frac{du}{n x^{n-1}} \).
Perform the substitution and simplify:
\[ \int \frac{\pi}{x \cdot u} \cdot \frac{du}{n x^{n-1}} = \frac{\pi}{n} \int \frac{1}{u} \cdot \frac{1}{x^n} \, du \]
Using the relation \( x^n = u + 1 \):
\[ \frac{\pi}{n} \int \frac{1}{u} \, du = \frac{\pi}{n} \log_e |u| + C \]
Substitute back \( u = x^n - 1 \):
\[ \frac{\pi}{n} \log_e |x^n - 1| + C \]