7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
23. ∫ 1/(25 - x²)^(3/2) dx

Verified step by step guidance

1. Step 1: Recognize the integral involves a term of the form \( a^2 - x^2 \), which suggests using the trigonometric substitution \( x = a \sin(\theta) \). Here, \( a = 5 \) because \( 25 = 5^2 \). Substitute \( x = 5 \sin(\theta) \), and compute \( dx = 5 \cos(\theta) d\theta \).
2. Step 2: Substitute \( x = 5 \sin(\theta) \) and \( dx = 5 \cos(\theta) d\theta \) into the integral. The term \( 25 - x^2 \) becomes \( 25 - 25 \sin^2(\theta) \), which simplifies to \( 25 \cos^2(\theta) \) using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
3. Step 3: Rewrite the integral in terms of \( \theta \): \( \int \frac{1}{(25 \cos^2(\theta))^{3/2}} \cdot 5 \cos(\theta) d\theta \). Simplify the denominator \( (25 \cos^2(\theta))^{3/2} \) to \( 125 \cos^3(\theta) \). The integral becomes \( \int \frac{5 \cos(\theta)}{125 \cos^3(\theta)} d\theta \).
4. Step 4: Simplify the fraction \( \frac{5 \cos(\theta)}{125 \cos^3(\theta)} \) to \( \frac{1}{25 \cos^2(\theta)} \). The integral now becomes \( \int \frac{1}{25 \cos^2(\theta)} d\theta \), which can be rewritten as \( \frac{1}{25} \int \sec^2(\theta) d\theta \).
5. Step 5: Evaluate \( \int \sec^2(\theta) d\theta \), which is a standard integral equal to \( \tan(\theta) \). After finding \( \tan(\theta) \), use the substitution \( \theta = \arcsin(x/5) \) to express the result back in terms of \( x \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² - x²), √(x² + a²), or √(x² - a²).

Integral of Trigonometric Functions

Understanding the integrals of trigonometric functions is essential for evaluating integrals after substitution. Common integrals include ∫ sin(θ) dθ = -cos(θ) + C and ∫ cos(θ) dθ = sin(θ) + C. Familiarity with these integrals allows for the effective evaluation of the transformed integral, leading to the final solution after reverting back to the original variable.

Pythagorean Identity

The Pythagorean identity, sin²(θ) + cos²(θ) = 1, is a fundamental relationship in trigonometry that is often used in conjunction with trigonometric substitution. This identity helps simplify expressions involving trigonometric functions, especially when converting back to the original variable after integration. Recognizing how to manipulate this identity is crucial for solving integrals that arise from trigonometric substitutions.