Problem 10 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-\infty}^{2} \frac{2 d x}{x^{2}+4} $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{3\pi}{4} \).

Step 1: Recognize the Integral Type

The given integral is a definite integral of the form \( \int \frac{a}{x^2 + b^2} \ dx \), which can generally be solved using a trigonometric substitution. This form resembles the derivative of the arctangent function.

Step 2: Apply Trigonometric Substitution

We recognize the integrand \( \frac{2}{x^2 + 4} \) as a function suitable for substitution with \( x = 2\tan(\theta) \). This leads to \( dx = 2\sec^2(\theta) \ d\theta \). Substituting these into the integral gives:\[\int \frac{2}{(2\tan(\theta))^2 + 4} \cdot 2\sec^2(\theta) \ d\theta = \int \frac{4\sec^2(\theta)}{4\tan^2(\theta) + 4} \ d\theta\]

Step 3: Simplify the Integral

The integral simplifies to:\[\int \frac{\sec^2(\theta)}{\tan^2(\theta) + 1} \ d\theta\]Since \( \tan^2(\theta) + 1 = \sec^2(\theta) \), the integral becomes:\[\int d\theta = \theta + C\]

Step 4: Convert Back to x

Since \( x = 2\tan(\theta) \), we have \( \theta = \tan^{-1}\left(\frac{x}{2}\right) \). Thus, our indefinite integral becomes:\[\tan^{-1}\left(\frac{x}{2}\right) + C\]

Step 5: Evaluate Definite Integral

Using the limits of integration from \(-\infty\) to 2, we substitute into the antiderivative:\[\left[ \tan^{-1}\left(\frac{x}{2}\right) \right]_{-\infty}^{2}\]This equals:\[\tan^{-1}(1) - \lim_{x \to -\infty} \tan^{-1}\left(\frac{x}{2}\right)\]\( \tan^{-1}(1) = \frac{\pi}{4} \) and \( \lim_{x \to -\infty} \tan^{-1}\left(\frac{x}{2}\right) = -\frac{\pi}{2} \), resulting in:\[\frac{\pi}{4} - \left(-\frac{\pi}{2}\right) = \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4}\]

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify integrals, especially those involving quadratic expressions under radicals or in denominators. For the integral \( \int \frac{2}{x^2 + 4} \, dx \), the substitution choice helps in transforming a complex looking integral into a simpler form.

In this case, we substitute \( x = 2\tan(\theta) \). Why? Because \( x^2 + 4 \) matches the identity \( a^2 + b^2\tan^2(\theta) \). This form suggests substituting with the tangent function due to the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Here's what you do:

• Express \( x \) using the trigonometric function, i.e., \( x = 2\tan(\theta) \).
• Compute the derivative \( dx = 2\sec^2(\theta) \, d\theta \).
• Replace \( x \) and \( dx \) in the integral, transforming it into a trigonometric integral.

This substitution results in simpler algebra, making the integration process more straightforward.

Arctangent Function

The arctangent function, \( \tan^{-1}(x) \), is closely linked to integrals of the form \( \int \frac{a}{x^2 + b^2} \, dx \). Recognizing that the derivative \( \frac{d}{dx} \tan^{-1}(x) = \frac{1}{1 + x^2} \), can simplify integration tasks drastically.

In this solution, once we've simplified the integral to \( \int \frac{\sec^2(\theta)}{\sec^2(\theta)} \, d\theta \), it directly simplifies to \( \int d\theta \), resulting in \( \theta \).

To express \( \theta \) back in terms of \( x \), recall the original substitution: \( x = 2\tan(\theta) \). Thus, \( \theta = \tan^{-1}\left(\frac{x}{2}\right) \).

This conversion is essential, enabling the solution to revert the variable of integration from \( \theta \) back to \( x \), yielding the antiderivative expression. Understanding this link between trigonometric substitution and the arctangent function streamlines solving these types of integrals.

Limit Evaluation

Limit evaluation is a critical step in definite integrals, especially those involving infinite bounds like \( \int_{-\infty}^{2} \). Evaluating such limits ensures the correctness of the solution when calculating an integral with infinity as one of its limits.

For the given problem, we evaluate \( \left[ \tan^{-1}\left(\frac{x}{2}\right) \right]_{-\infty}^{2} \). When solving, each bound is substituted into the antiderivative:

• At the upper limit, \( x = 2 \), compute \( \tan^{-1}(1) \), leading to \( \frac{\pi}{4} \).
• As \( x \to -\infty \), calculate the limit of \( \tan^{-1}\left(\frac{x}{2}\right) \). Since \( \frac{x}{2} \to -\infty \), \( \tan^{-1}(-\infty) = -\frac{\pi}{2} \).

The definite integral then results in calculating the difference: \( \frac{\pi}{4} - (-\frac{\pi}{2}) \), simplifying to \( \frac{3\pi}{4} \).

Understanding limit evaluation integrates prose and math, guaranteeing the solution accurately reflects changes as \( x \) approaches infinity. This ensures proper computation of areas under curves extending infinitely in one direction.