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

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

Answer 2

Begin with the integral: 

\[ \int \frac{\pi}{x^{n+1} - x} \, dx \]

Step 1: Factor the denominator:

We can factor the denominator as follows:

\[ x^{n+1} - x = x \cdot (x^n - 1) \] Thus, the integral becomes: \[ \int \frac{\pi}{x \cdot (x^n - 1)} \, dx \]

Step 2: Use substitution:

Let \( u = x^n - 1 \), so \( du = n x^{n-1} \, dx \). This gives \( dx = \frac{du}{n x^{n-1}} \).

Step 3: Substitute \( u \) and simplify:

\[ \int \frac{\pi}{x \cdot u} \cdot \frac{du}{n x^{n-1}} = \frac{\pi}{n} \int \frac{1}{u} \cdot \frac{1}{x^n} \, du \]

Step 4: Simplify the expression further:

Since \( x^n = u + 1 \), we substitute it into the expression: \[ \frac{\pi}{n} \int \frac{1}{u} \, du \]

Step 5: Integrate:

The integral of \( \frac{1}{u} \) is \( \log_e |u| \), so we get: \[ \frac{\pi}{n} \log_e |u| + C \]

Step 6: Substitute back for \( u \):

Substituting \( u = x^n - 1 \) back into the result, we get: \[ \frac{\pi}{n} \log_e |x^n - 1| + C \]