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Chapter 7: Problem 11

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\infty} e^{-2 x} d x$$

Short Answer

Expert verified
Answer: The integral of the function \(e^{-2x}\) from \(0\) to \(\infty\) is \(\frac{1}{2}\).

Step by step solution

01

Determine if the integral converges or diverges

To determine if the given integral converges or diverges, we need to find the limit of integration as \(x\) approaches infinity, which means: $$\lim_{R \to \infty} \int_{0}^{R} e^{-2x}dx$$
02

Calculate the antiderivative

Let's find the antiderivative of the given function, for this we use the substitution method: let \(u = -2x\), therefore, \(\frac{du}{dx} = -2 \implies dx = \frac{-1}{2} du\) $$\int e^{-2x} dx = \int e^u \left(\frac{-1}{2} du\right) = \frac{-1}{2} \int e^u du$$ Now, integrate as usual: $$\frac{-1}{2} \int e^u du = \frac{-1}{2} e^u+ C= \frac{-1}{2} e^{-2x} + C$$
03

Evaluate the definite integral

We substitute the limits of integration \(0\) and \(R\) and find the limit as \(R\) approaches infinity: $$\lim_{R \to \infty} \left[\frac{-1}{2} e^{-2x} \right]_0^R = \lim_{R \to \infty} \left(\frac{-1}{2}e^{-2R} - \frac{-1}{2}e^0\right) = \lim_{R \to \infty} \left(\frac{-1}{2}e^{-2R} + \frac{1}{2}\right)$$ As \(R\) approaches infinity, \(e^{-2R}\) approaches \(0\). Thus, $$\lim_{R \to \infty} \left(\frac{-1}{2}e^{-2R} + \frac{1}{2}\right) = 0 + \frac{1}{2}$$
04

Final answer

The given integral converges, and the solution is: $$\int_{0}^{\infty} e^{-2 x} d x = \frac{1}{2}$$

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Most popular questions from this chapter

A remarkable integral It is a fact that \(\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{m} x}=\frac{\pi}{4}\) for all real numbers \(m .\) a. Graph the integrand for \(m=-2,-3 / 2,-1,-1 / 2,0,1 / 2\) \(1,3 / 2,\) and \(2,\) and explain geometrically how the area under the curve on the interval \([0, \pi / 2]\) remains constant as \(m\) varies. b. Use a computer algebra system to confirm that the integral is constant for all \(m.\)

Another Simpson's Rule formula is \(S(2 n)=\frac{2 M(n)+T(n)}{3},\) for \(n \geq 1 .\) Use this rule to estimate \(\int_{1}^{e} 1 / x d x\) using \(n=10\) subintervals.

A differential equation and its direction field are given. Sketch a graph of the solution that results with each initial condition. $$\begin{aligned}&y^{\prime}(t)=\frac{\sin t}{y},\\\&y(-2)=-2 \text { and }\\\&y(-2)=2\end{aligned}$$

Use symmetry to evaluate the following integrals. a. \(\int_{-\infty}^{\infty} e^{-|x|} d x\) b. \(\int_{-\infty}^{\infty} \frac{x^{3}}{1+x^{8}} d x\)

Solve the following problems using the method of your choice. $$w^{\prime}(t)=2 t \cos ^{2} w, w(0)=\pi / 4$$
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