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Chapter 7: Problem 10

All the integrals in problem are improper and converge. Explain in each case why the integral is improper, and evaluate each integral. $$ \int_{-\infty}^{\infty} x^{3} e^{-x^{4}} d x $$

Short Answer

Expert verified
This integral is improper due to infinite limits and evaluates to 0 because the function is odd and symmetric.

Step by step solution

01

Identify the Improprieties

The integral \( \int_{-\infty}^{\infty} x^{3} e^{-x^{4}} \, dx \) is improper due to infinite limits of integration: from \(-\infty\) to \(\infty\). This is what makes the integral improper.
02

Rewrite the Integral with Limits

To evaluate the improper integral, we rewrite it as a limit: \[\int_{-\infty}^{\infty} x^{3} e^{-x^{4}} \, dx = \lim_{a \to -\infty, b \to \infty} \int_{a}^{b} x^{3} e^{-x^{4}} \, dx.\]
03

Evaluate the Integral Symmetry

Note that \(x^3\) is an odd function while \(e^{-x^4}\) is an even function, making their product \((x^3 e^{-x^4})\) an odd function. The integral of an odd function over symmetric limits \(-a\) to \(a\) yields zero. Therefore, \[\int_{-\infty}^{\infty} x^{3} e^{-x^{4}} \, dx = 0.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Limits of Integration
Improper integrals often involve infinite limits of integration. This simply means that the bounds within which you are integrating extend to infinity, like
  • from \(-\infty\) to a finite number,
  • from a finite number to \(\infty\), or
  • from \(-\infty\) to \(\infty\).
In this problem, we're dealing with the third case, where both bounds are infinite.

To evaluate such an integral, we express it as a limit. We often rewrite it in the form \[\lim_{a \to -\infty, b \to \infty} \int_{a}^{b} f(x) \, dx.\] This approach allows us to treat the problem of infinite bounds in a manageable way. By calculating the integral over a finite range and then extending that range to infinity, we can determine whether the integral converges to a finite value or not.
Odd and Even Functions
When considering functions like in the problem \(x^3 e^{-x^4}\), understanding odd and even functions can greatly simplify the evaluation.

  • Odd function: A function \(f(x)\) is odd if \(f(-x) = -f(x)\). An example is \(x^3\).
  • Even function: A function \(g(x)\) is even if \(g(-x) = g(x)\). An example is \(e^{-x^4}\).
When you multiply an odd function by an even function, the result is typically an odd function. This is the case with the product \(x^3 e^{-x^4}\).

Because the product is an odd function, integrating it over a symmetric interval from \(-a\) to \(a\) will result in zero. This is due to the property of odd functions that the areas under the curve on either side of the y-axis cancel each other out.
Evaluating Integrals
Evaluating integrals, particularly improper ones, involves several steps and techniques. In this case, we used the symmetry properties of the function being integrated.

Given the integral \[\int_{-\infty}^{\infty} x^{3} e^{-x^{4}} \, dx,\]the evaluation hinges on recognizing the integrand as an odd function.

There鈥檚 no need to find an antiderivative or perform complex calculations because the symmetry of odd functions over symmetric intervals simplifies the problem significantly.

This integral evaluates to zero directly because the negative area on one side of the y-axis cancels out with the positive area on the other side. Using symmetry is a powerful tool in calculus to simplify calculations and reach conclusions quickly.

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Most popular questions from this chapter

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x}} d x $$

(a) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{k} R $$ for \(R\) close to 0 . (b) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{2}+\frac{a}{4 k}(R-k) $$ for \(R\) close to \(k\).

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\cos x, x=1, \text { error }<10^{-2} $$

(a) Show that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$$ (b) Use your result in (a) to show that, for any \(c>0\), $$c x \geq \ln x$$ for sufficiently large \(x\).46. (a) Show that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$$ (b) Use your result in (a) to show that, for any \(c>0\), $$c x \geq \ln x$$ for sufficiently large \(x\). (c) Use your result in (b) to show that, for any \(p>0\), $$x^{p} e^{-x} \leq e^{-x / 2}$$ provided that \(x\) is sufficiently large. (d) Use your result in (c) to show that, for any \(p>0\), $$\int_{0}^{\infty} x^{p} e^{-x} d x$$ is convergent.

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=x^{1 / 5}, a=1, n=3 ; x=0.9 $$
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