Math Problem Statement

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

Solution

The integral in the image you provided is:

01xn11+x2dx\int_{0}^{1} \frac{x^{n-1}}{1 + x^2} \, dx

This is a definite integral with limits from 00 to 11. The integrand involves a power of xx and a rational function.

Steps to Solve:

  1. Substitution Method: Consider using the substitution x=tan(θ)x = \tan(\theta), which simplifies the expression 1+x21 + x^2 as sec2(θ)\sec^2(\theta). This substitution is often useful for integrals involving 1+x21 + x^2 in the denominator.

  2. Substitute: Let x=tan(θ)x = \tan(\theta), then dx=sec2(θ)dθdx = \sec^2(\theta) \, d\theta. The limits of integration will change accordingly:

    • When x=0x = 0, θ=0\theta = 0
    • When x=1x = 1, θ=π4\theta = \frac{\pi}{4}

    The integral becomes: 0π4tann1(θ)sec2(θ)sec2(θ)dθ=0π4tann1(θ)dθ\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 nn. For specific values of nn, 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 nn, 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 xx and rational functions?
  2. What are some typical substitutions used in integrals involving 1+x21 + x^2?
  3. How does the value of nn 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(θ)x = \tan(\theta) generally simplify integrals?

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

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Math Problem Analysis

Mathematical Concepts

Definite Integral
Substitution Method
Trigonometric Substitution

Formulas

\(x = \tan(\theta)\)
\( dx = \sec^2(\theta) \, d\theta\)

Theorems

Integration by Substitution
Trigonometric Identities

Suitable Grade Level

University Level