Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \operatorname{sech} x d x\)

Step-by-Step Solution

Answer

The integral converges to \( \pi \).

Step 1: Understand the Function

The function to be integrated is the hyperbolic secant function, denoted as \( \operatorname{sech} x = \frac{1}{\cosh x} \), where \( \cosh x = \frac{e^x + e^{-x}}{2} \). This function is even, meaning \( \operatorname{sech}(-x) = \operatorname{sech}(x) \). Knowing it is even will help us simplify the integral.

Step 2: Setup the Integral for Evaluation

Since the hyperbolic secant is an even function, we can use symmetry to write the integral as: \[ \int_{-\infty}^{\infty} \operatorname{sech} x \, dx = 2 \int_{0}^{\infty} \operatorname{sech} x \, dx. \] This is because the integral from \( -\infty \) to 0 is equal to the integral from 0 to \( \infty \).

Step 3: Check for Convergence

To determine if \( \int_{0}^{\infty} \operatorname{sech} x \, dx \) converges, we need to analyze the behavior of \( \operatorname{sech} x \) as \( x \to \infty \). Note that \( \operatorname{sech} x = \frac{2}{e^x + e^{-x}} \approx 2e^{-x} \) as \( x \to \infty \). This resembles the exponential decay \( \int_{0}^{\infty} e^{-x} \, dx \), which converges.

Step 4: Evaluate the Convergent Integral

The integral \( \int_{0}^{\infty} \operatorname{sech} x \, dx \) can be approached using a substitution. Let \( I = \int_{0}^{\infty} \operatorname{sech} x \, dx \). We evaluate and find that using substitution \( I = \int_{0}^{\infty} \frac{2}{e^x + e^{-x}} \, dx = \pi. \)

Step 5: Calculating the Final Result

Now, using the symmetry property: \[ \int_{-\infty}^{\infty} \operatorname{sech} x \, dx = 2 \times \frac{\pi}{2} = \pi. \] Therefore, the value of the improper integral is \( \pi \).

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are a vital part of calculus, especially when dealing with integrals. These functions are analogous to the trigonometric functions but are built on hyperbolas instead of circles. A key hyperbolic function found in this exercise is the hyperbolic secant function, denoted as \( \operatorname{sech} x \), which equals \( \frac{1}{\cosh x} \). Here, \( \cosh x \) is defined as \( \frac{e^x + e^{-x}}{2} \).

Hyperbolic functions have properties that are similar to their trigonometric counterparts. For instance, \( \operatorname{sech} x \) is even, meaning its value is the same for both \( x \) and \( -x \). Recognizing this even property allows us to simplify certain calculations by focusing only on half of the domain, commonly from 0 to \( \infty \), while doubling the result later to account for the symmetry.

Understanding these properties helps one handle integrals and other calculus problems involving hyperbolic functions more effectively.

Explaining Convergence in Integrals

Convergence is a critical concept when evaluating improper integrals. An integral is said to converge if it results in a finite number. If we obtain an infinite result, the integral diverges. In this exercise, we encounter an improper integral that spans from \( -\infty \) to \( \infty \).

In such cases, assessing convergence is crucial before proceeding to solve the integral. For \( \int_{0}^{\infty} \operatorname{sech} x \, dx \), the function \( \operatorname{sech} x \) behaves similarly to \( 2e^{-x} \) as \( x \to \infty \), which is known to converge. \( e^{-x} \) is a classic example of exponential decay, reducing to zero very fast.

By comparing \( \operatorname{sech} x \) with \( e^{-x} \), we use a comparison test to determine convergence. This approach simplifies judging the behavior of the function at infinity, a key step in proving an improper integral's convergence.

Strategies for Calculus Problem Solving

Solving calculus problems, especially those involving improper integrals, often requires strategic thinking. Here, we have a problem that involves evaluating an integral over an infinite range, thus requiring us to verify its convergence initially.

In this context, using symmetry is a powerful strategy. For even functions like \( \operatorname{sech} x \), evaluate half the domain and double the result to simplify computations. This reduces the workload and focuses only on a manageable range.

• Determine if the function is even or odd to utilize properties of symmetry.
• Split the integral based on the symmetry to simplify.
• Use substitution to evaluate the integral efficiently.
• Compare the function's behavior with simpler, well-known functions to establish convergence.

These steps make tackling calculus problems with infinite limits far more manageable, ensuring an accurate solution with a clear thoughtful methodology.