Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1

Verified step by step guidance

1. Recognize that the integral has the form \( \int \frac{dx}{(a^2 - x^2)^{3/2}} \) with \( a = 2 \). This suggests using a trigonometric substitution to simplify the integrand.
2. Use the substitution \( x = 2 \sin\theta \), which implies \( dx = 2 \cos\theta \, d\theta \). This substitution transforms the expression under the square root: \( 4 - x^2 = 4 - 4\sin^2\theta = 4\cos^2\theta \).
3. Rewrite the integral in terms of \( \theta \): replace \( dx \) and \( (4 - x^2)^{3/2} \) with their trigonometric equivalents, resulting in an integral involving powers of \( \cos\theta \).
4. Adjust the limits of integration to match the substitution: when \( x = 0 \), find \( \theta \); when \( x = 1 \), find the corresponding \( \theta \).
5. Simplify the integral and evaluate it with respect to \( \theta \). After integration, substitute back to express the answer in terms of \( x \) if needed.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the integral simplifies using trigonometric identities, making it easier to solve.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two points. When using substitution methods, it is important to adjust the limits of integration to match the new variable or revert to the original variable after integration to evaluate the integral correctly.

Integration of Rational Functions Involving Powers

Integrals involving rational functions with powers, such as (4 - x²)^(-3/2), often require recognizing patterns or applying substitution methods. Understanding how to manipulate and integrate these expressions is essential for solving such integrals efficiently.