https://file.aiquickdraw.com/mathbot/file/17253782936669at2ij8i.jpg

The integral in the image you provided is:

\[
\int_{0}^{1} \frac{x^{n-1}}{1 + x^2} \, dx
\]

This is a definite integral with limits from \(0\) to \(1\). The integrand involves a power of \(x\) and a rational function.

### Steps to Solve:
1. **Substitution Method**: Consider using the substitution \( x = \tan(\theta) \), which simplifies the expression \( 1 + x^2 \) as \( \sec^2(\theta) \). This substitution is often useful for integrals involving \( 1 + x^2 \) in the denominator.

2. **Substitute**: Let \( x = \tan(\theta) \), then \( dx = \sec^2(\theta) \, d\theta \). The limits of integration will change accordingly:
   - When \( x = 0 \), \( \theta = 0 \)
   - When \( x = 1 \), \( \theta = \frac{\pi}{4} \)

   The integral becomes:
   \[
   \int_{0}^{\frac{\pi}{4}} \frac{\tan^{n-1}(\theta) \cdot \sec^2(\theta)}{\sec^2(\theta)} d\theta = \int_{0}^{\frac{\pi}{4}} \tan^{n-1}(\theta) \, d\theta
   \]

3. **Evaluate**: This resulting integral can be evaluated depending on the value of \(n\). For specific values of \(n\), the integral might simplify, or you might use special functions such as the Beta or Gamma functions to solve it.

Would you like further details on how to proceed with evaluating this integral for a specific \(n\), or any other questions on related topics? 

Here are 5 related questions to explore:
1. How can the Beta function be used to evaluate integrals involving powers of \(x\) and rational functions?
2. What are some typical substitutions used in integrals involving \(1 + x^2\)?
3. How does the value of \(n\) affect the difficulty or method for solving this integral?
4. Can this integral be solved using series expansion techniques?
5. How does the substitution \(x = \tan(\theta)\) generally simplify integrals?

**Tip**: Always check if the integral limits change when performing substitution; this helps avoid mistakes in definite integrals.

Evaluate the definite integral: \(\int_{0}^{1} \frac{x^{n-1}}{1 + x^2} \, dx\)

Learn how to evaluate the definite integral \(\int_{0}^{1} \frac{x^{n-1}}{1 + x^2} \, dx\) using trigonometric substitution. This problem covers key concepts like the substitution method, trigonometric identities, and evaluating integrals with changing limits. Suitable for university-level students.

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