Problem 10 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Rewrite the Integral with Limits

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

Step 3: 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, \[\int_{-\infty}^{\infty} x^{3} e^{-x^{4}} \, dx = 0.\]

Key Concepts

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’s 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.