2 **Title: Evaluating an Integral Using Trigonometric Substitution****Problem Statement:**Use a trigonometric substitution to evaluate the integral:\[ \int \frac{dx}{(x^2 + 4)^2} \] **Solution Outline:**To solve the integral \(\int \frac{dx}{(x^2 + 4)^2}\) using trigonometric substitution, follow these steps:1. **Identify the Trigonometric Substitution:** - For integrals involving \(\sqrt{x^2 + a^2}\), use the substitution \(x = a \tan(\theta)\). - In this problem, \(x^2 + 4\) fits the form with \(a^2 = 4\) (i.e., \(a = 2\)). - Hence, we use the substitution \(x = 2 \tan(\theta)\).2. **Calculate the Differential (dx):** - With \(x = 2 \tan(\theta)\), \(dx = 2 \sec^2(\theta) d\theta\).3. **Substitute into the Integral:** - Substituting \(x = 2 \tan(\theta)\) and \(dx = 2 \sec^2(\theta) d\theta\) into the integral: \[ \int \frac{2 \sec^2(\theta) d\theta}{(4 \tan^2(\theta) + 4)^2} \] - Simplify the denominator: \[ 4 \tan^2(\theta) + 4 = 4 (\tan^2(\theta) + 1) = 4 \sec^2(\theta) \] - Therefore, the integral becomes: \[ \int \frac{2 \sec^2(\theta) d\theta}{(4 \sec^2(\theta))^2} = \int \frac{2 \sec^2(\theta) d\theta}{16 \sec^4(\theta)} \] - Simplify further: \[ \int \frac{2 \sec^2(\theta) d\theta}{16 \sec^4(\theta)} = \int \frac{2 d\theta}{16 \sec^2(\theta)} = \frac{1}{8}

Name: 2 **Title: Evaluating an Integral Using Trigonometric Substitution****
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: 2 **Title: Evaluating an Integral Using Trigonometric Substitution****Problem Statement:**Use a trigonometric substitution to evaluate the integral:\[ \int \frac{dx}{(x^2 + 4)^2} \] **Solution Outline:**To solve the integral \(\int \frac{dx}{(x^2 + 4