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Chapter 6: Problem 17

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} x^{2} e^{-x} d x $$

Short Answer

Expert verified
The improper integral \(\int_{0}^{\infty} x^{2} e^{-x} dx\) converges and its value is 2.

Step by step solution

01

Verify Convergence

To determine whether the given integral converges or diverges, compare it with a known convergent integral. We know that, in general, the exponential function decays faster than any power of \(x\). Thus, the integral \(\int_{0}^{\infty} x^{2} e^{-x} dx\) converges because the \(e^{-x}\) term, as \(x\) approaches infinity, reduces faster than \(x^{2}\) can increase.
02

Integrate Using the IBP Method (First Time)

To evaluate the integral, use integration by parts (IBP). Set \(u = x^2\) and \(dv = e^{-x} dx\). So, \(du = 2x dx\) and \(v = -e^{-x}\). Using the formula for IBP (\(\int u dv = uv - \int v du\)), the integral transforms into:\[\int_{0}^{\infty} x^{2} e^{-x} dx = -x^{2} e^{-x} \Biggr|_{0}^{\infty} + \int_{0}^{\infty} 2x e^{-x} dx.\]
03

Apply Limits to the First Term

We first evaluate the term \(-x^2e^{-x}\) from 0 to infinity, which gives us 0. Hence, the integral reduces to: \[\int_{0}^{\infty} 2x e^{-x} dx.\]
04

Integrate Using the IBP Method (Second Time)

We still have an integral we need to evaluate. We again use IBP. This time we set \(u = x\) and \(dv = 2e^{-x} dx\). We then find \(du = dx\) and \(v = -2e^{-x}\). Using the IBP formula again, the integral transforms into:\[\int_{0}^{\infty} 2x e^{-x} dx = -2x e^{-x} \Biggr|_{0}^{\infty} + 2 \int_{0}^{\infty} e^{-x} dx.\]
05

Evaluate the Remaining Integral and Apply Limits

We now need to evaluate the integral \(\int_{0}^{\infty} e^{-x} dx\), which is simply \(-e^{-x}\) from 0 to infinity. Evaluating the limits and summing all terms, we find that the original improper integral, \(\int_{0}^{\infty} x^{2} e^{-x} dx\), is equal to 2.

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Most popular questions from this chapter

(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta $$

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cos a t $$
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