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Chapter 7: Problem 20

Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+2 x+5}$$

Short Answer

Expert verified
If it converges, find its value. Answer: The given integral converges with a value of \(\frac{\pi}{2}\).

Step by step solution

01

Complete the square in the denominator

First, we need to complete the square of the polynomial in the denominator. The denominator is given by \(x^2 + 2x + 5\). In order to complete the square, we add and subtract the square of half the coefficient of x, which is \((2/2)^2 = 1\). Therefore: $$x^2 + 2x + 5 = (x^2 + 2x + 1) + 4 = (x + 1)^2 + 4$$ Now the integrand becomes: $$\frac{1}{(x + 1)^2 + 4}$$
02

Apply the substitution

To turn our integrand into a more familiar form, we can perform the following substitution: Let \(u = x + 1\). Then, \(du = dx\) and our integral becomes: $$\int_{-\infty}^{\infty} \frac{1}{u^2 + 4} du$$
03

Rewrite the integrand

Next, rewrite the integrand by factoring out the constant term "4": $$\frac{1}{u^2+4} = \frac{1}{4}\frac{1}{\frac{u^2}{4}+1}$$
04

Use another substitution

Use the following substitution to simplify even further: Let \(v = \frac{u}{2}\), then \(dv = \frac{1}{2} du\), and \(2 dv = du\). The integral becomes: $$\frac{1}{4} \int_{-\infty}^{\infty} \frac{2 dv}{v^2 + 1}$$
05

Apply the arctangent integral formula

Now we have an integral in the recognizable form for the arctangent integral formula: $$\int \frac{1}{v^2 + 1} dv = \arctan{v} + C$$ Now we can solve the integral: $$\frac{1}{4} \int_{-\infty}^{\infty} \frac{2 dv}{v^2 + 1} = \frac{1}{2} \left. \arctan{v} \right|_{-\infty}^{\infty}$$
06

Evaluate the integral and determine convergence

Now we simply evaluate the arctangent function at infinity and negative infinity: $$\frac{1}{2} \left( \lim_{v \to \infty} \arctan{v} - \lim_{v \to -\infty} \arctan{v} \right) = \frac{1}{2} (\frac{\pi}{2} - \frac{-\pi}{2}) = \boxed{\frac{\pi}{2}}$$ Thus, the integral converges and the value is \(\frac{\pi}{2}\).

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Most popular questions from this chapter

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \tan ^{4} 3 y d y$$

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Determine whether the following statements are true and give an explanation or counterexample. a. If \(f\) is continuous and \(0p\) d. If \(\int_{1}^{\infty} x^{-p} d x\) exists, then \(\int_{1}^{\infty} x^{-q} d x\) exists, where \(q>p\) e. \(\int_{1}^{\infty} \frac{d x}{x^{3 p+2}}\) exists, for \(p>-\frac{1}{3}\)

Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.

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