7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx

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1. Step 1: Recognize that the integral involves a term of the form \( \sqrt{a^2 - x^2} \), which suggests using the trigonometric substitution \( x = a \sin(\theta) \). Here, \( a = 10 \) because \( 100 = 10^2 \). Substitute \( x = 10 \sin(\theta) \), so \( dx = 10 \cos(\theta) d\theta \).
2. Step 2: Substitute \( x = 10 \sin(\theta) \) and \( dx = 10 \cos(\theta) d\theta \) into the integral. Replace \( \sqrt{100 - x^2} \) with \( \sqrt{100 - 100 \sin^2(\theta)} = \sqrt{100 \cos^2(\theta)} = 10 \cos(\theta) \). The integral becomes \( \int \frac{(10 \sin(\theta))^2}{(10 \cos(\theta))^3} \cdot 10 \cos(\theta) d\theta \).
3. Step 3: Simplify the expression. The numerator \( (10 \sin(\theta))^2 \) becomes \( 100 \sin^2(\theta) \), and the denominator \( (10 \cos(\theta))^3 \cdot 10 \cos(\theta) \) becomes \( 1000 \cos^4(\theta) \). The integral simplifies to \( \int \frac{100 \sin^2(\theta)}{1000 \cos^4(\theta)} d\theta \), which reduces further to \( \int \frac{\sin^2(\theta)}{10 \cos^4(\theta)} d\theta \).
4. Step 4: Use the Pythagorean identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \) to rewrite the integral. Substitute \( \sin^2(\theta) \) with \( 1 - \cos^2(\theta) \), resulting in \( \int \frac{1 - \cos^2(\theta)}{10 \cos^4(\theta)} d\theta \). Split the integral into two parts: \( \int \frac{1}{10 \cos^4(\theta)} d\theta - \int \frac{\cos^2(\theta)}{10 \cos^4(\theta)} d\theta \).
5. Step 5: Simplify each term. The first term becomes \( \int \frac{1}{10 \cos^4(\theta)} d\theta \), and the second term simplifies to \( \int \frac{1}{10 \cos^2(\theta)} d\theta \). These integrals can be solved using standard trigonometric integral techniques or identities. After solving, back-substitute \( \theta \) using \( x = 10 \sin(\theta) \) to express the result 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) or √(x² + a²).

Integral of a Function

The integral of a function represents the area under the curve of that function over a specified interval. In the context of the given problem, evaluating the integral ∫ x²/(100 - x²)^(3/2) dx involves finding the antiderivative of the function. Understanding how to compute integrals, including techniques like substitution and integration by parts, is essential for solving complex integrals.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is crucial when performing trigonometric substitutions, as it allows for the simplification of expressions involving trigonometric functions. By using this identity, one can often express the integral in terms of a single trigonometric function, making it easier to evaluate.