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Magazine Chapter 8: Problem 16
Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+a^{2}}, a>0$$
Short Answer
Expert verified
Answer: The value of the integral is $$\frac{\pi}{a}$$.
Step by step solution
01
Set up the integral
The given integral is:
$$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+a^{2}}$$
where \(a>0\).
02
Trigonometric substitution
Let:
$$x = a \tan \theta$$
Then:
$$dx = a\sec^2\theta d\theta$$
03
Transform the integral
Substituting \(x\) and \(dx\), we have:
$$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+a^{2}} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a\sec^2\theta d\theta}{(a\tan\theta)^2 + a^2}$$
04
Simplify the integral
Let's simplify the integral expression:
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a\sec^2\theta d\theta}{(a\tan\theta)^2 + a^2} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a\sec^2\theta d\theta}{a^2\tan^2\theta + a^2}$$
Combine the \(a^2\) terms in the denominator:
$$= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a\sec^2\theta d\theta}{a^2(\tan^2\theta + 1)}$$
We know that \(\sec^2\theta = \tan^2\theta + 1\), so:
$$= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a\sec^2\theta d\theta}{a^2\sec^2\theta}$$
Now, cancel out the common terms:
$$= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a d\theta}{a^2}$$
05
Compute the integral
Now, we can compute the integral:
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a d\theta}{a^2} = \frac{1}{a}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta$$
$$= \frac{1}{a}[\theta]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}$$
$$= \frac{1}{a}\left(\frac{\pi}{2} + \frac{\pi}{2}\right)$$
06
Simplify the final result
Finally, we simplify the result:
$$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+a^{2}} = \frac{\pi}{a}$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
When evaluating some integrals, particularly those involving square roots or quadratic expressions, trigonometric substitution is a valuable technique. It's a method aimed at simplifying integrals by replacing a variable with a trigonometric function. This is especially useful when the integral involves expressions like \(x^2 + a^2\), as in our exercise.
Here's how it works in our context:
Here's how it works in our context:
- For the integral \(\int \frac{dx}{x^2 + a^2}\), set \(x = a \tan\theta\) which implies that \(dx = a \sec^2\theta d\theta\).
- This substitution transforms the difficult integral into a form that's easier to work with using standard trigonometric identities.
- The key identity here is \(\tan^2\theta + 1 = \sec^2\theta\), which simplifies the integral considerably.
Convergence of Integrals
Before solving an integral, especially an improper one, it is crucial to determine if the integral converges, meaning it has a finite value, or if it diverges. An improper integral is characterized by having one or both limits approaching infinity or by the integrand becoming infinite at some point within the limits of integration.
For our exercise:
For our exercise:
- The integral \(\int_{-\infty}^{\infty} \frac{dx}{x^2 + a^2}\) is improper due to its limits of integration extending to infinity.
- We address convergence by examining the behavior of the integrand as \(x\) approaches \(-\infty\) and \(\infty\).
- The expression \(x^2 + a^2\) keeps the denominator positive and non-zero across its domain, ensuring no undefined behavior at any point.
- Thus, provided \(a > 0\), the integral converges, and we can proceed with evaluation.
Evaluating Integrals
With confirmation that our integral converges, we can move on to computing its value. Once an integral is simplified via a substitution or a reduction technique, like the trigonometric substitution employed here, the evaluation becomes more straightforward.
Here鈥檚 the process detailed:
Here鈥檚 the process detailed:
- After substitution, our integral reduces to \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{a d\theta}{a^2}\).
- This expression simplifies further to \(\frac{1}{a} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta\), breaking down to an integral of a constant.
- The final evaluation of this definite integral involves calculating the difference \([\theta]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\), which results in \(\pi\).
- We then multiply by \(\frac{1}{a}\), yielding the final result \(\frac{\pi}{a}\).
