Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+10} d x\)

Step-by-Step Solution

Answer

The integral converges to \( \frac{\pi}{3} \).

Step 1: Identify the Function

The function we need to integrate is \( f(x) = \frac{1}{x^2 + 2x + 10} \). This is a rational function, and we need to determine whether it converges when integrated from \(-\infty\) to \(\infty\).

Step 2: Identify the Nature of the Improper Integral

The given integral is of the form \( \int_{-\infty}^{\infty} f(x) \, dx \). To evaluate it, we must split it into two parts: \( \int_{-\infty}^{a} f(x) \, dx \) and \( \int_{a}^{\infty} f(x) \, dx \) where \( a \) is any real number (often chosen for convenience, such as 0).

Step 3: Simplify the Denominator

The denominator \( x^2 + 2x + 10 \) can be rewritten by completing the square: \( x^2 + 2x + 10 = (x+1)^2 + 9 \). Thus, the function becomes \( f(x) = \frac{1}{(x+1)^2 + 9} \).

Step 4: Recognize Form and Use Substitution

Recognize the form \( \frac{1}{(x-a)^2 + b^2} \) which resembles the derivative of the arctangent. We'll use the substitution \( u = x + 1 \) so that \( du = dx \) to simplify the computation.

Step 5: Change of Variables

Perform the substitution: \( x+1 = u \), so \( dx = du \). The integral bounds change accordingly as the variable \( u \) shifts by 1. Thus, we find \( \int \frac{1}{u^2 + 9} du \) from \( -\infty+1 \) to \( \infty+1 \).

Step 6: Evaluate the Integral

The expression \( \int \frac{1}{u^2 + 9} \, du \) can be identified with the standard integral form \( \int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \), where \( a = 3 \). Thus, compute the value as \( \frac{1}{3} \left[ \arctan\left(\frac{u}{3}\right) \right]_{-\infty+1}^{\infty+1} \).

Step 7: Compute the Limits

We calculate \( \lim_{u \to \infty+1} \frac{1}{3} \arctan\left(\frac{u}{3}\right) - \lim_{u \to -\infty+1} \frac{1}{3} \arctan\left(\frac{u}{3}\right) \), which evaluates to \( \frac{1}{3} \left( \frac{\pi}{2} - (-\frac{\pi}{2}) \right) = \frac{1}{3} (\pi) = \frac{\pi}{3} \).

Step 8: Concluding the Integral

Since both limits exist and are finite, the integral converges and its value is \( \frac{\pi}{3} \).

Key Concepts

Convergence of Integrals

When dealing with improper integrals, like the one we have here, it's important to discuss the concept of convergence. An integral converges if its limit exists and is finite as you extend the interval to infinity of negative infinity. In our case with the integral \[\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+10} \, dx,\]we need to determine if evaluating it over an infinite range yields a finite result.
To check for convergence, break the integral into two sections:

• From \(-\infty\) to a chosen midpoint.
• From that midpoint to \(\infty\).

This separation helps verify convergence in parts. Each part needs to be evaluated to see if it approaches a finite number. If it does, then the entire integral converges. Otherwise, the integral diverges.

Rational Functions

A rational function is a ratio of two polynomials. It takes the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. In this exercise, \( f(x) = \frac{1}{x^2 + 2x + 10} \) is a rational function since it's expressed as \( \frac{1}{Q(x)} \) with
\( Q(x) = x^2 + 2x + 10 \). This constant numerator creates a unique pattern easily integrated further. To work with this function, it helps to simplify the denominator by completing the square. By doing so, you transform \( x^2 + 2x + 10 \) into \( (x+1)^2 + 9 \), making the function easier to manage.Understanding how to simplify rational functions is crucial before attempting integration, especially with improper integrals.

Integration Techniques

Integration can be tricky, especially with improper integrals spanning infinite intervals. Different techniques can assist in simplifying and evaluating them. Here, recognizing patterns is essential.When faced with integrals from\(-\infty\)to \(\infty\), applying integration techniques such as substitution methods becomes useful. Similarly, utilizing known integral forms helps. For example:
If you identify a component of your function as similar to a standard integral form like \( \int \frac{1}{u^2 + a^2} \, du \), you can leverage properties of known integral solutions, simplifying your work.

Substitution Method

Using substitution is a powerful technique in integration, especially for simplifying complex expressions into manageable ones. In this example, the substitution \( u = x + 1 \) helped streamline the calculation process. Here is how it works:
1. **Change Variables:** Identify a part of the integrand that, if replaced, simplifies the expression. In this case, \( u = x + 1 \) transforms \( x^2 + 2x + 10 \) into \( u^2 + 9 \).2. **Adjust the Limits:** When changing variables, remember that limits of integration also need adjusting to the new variable.3. **Integrate Simplified Expression:** Once substituted, the new integral becomes easier to evaluate using standard results. For example,\[\int \frac{1}{u^2 + 9} \, du\]resembles the formula \( \int \frac{1}{u^2+a^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C\).4. **Re-substitute:** Finally, return to your initial variable terms to express the integral in original terms.The substitution method not only simplifies the problem but keeps calculations clear, making it one of the efficient integration techniques to master in calculus.