Problem 42 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral \( \int_{-\infty}^{\infty} \frac{1}{e^x + e^{-x}} \, dx \) is convergent.

Step 1: Simplify the Integrand

The given integrand is \( \frac{1}{e^x + e^{-x}} \). This can be rewritten using hyperbolic cosine as \( \frac{2}{e^x + e^{-x}} = \frac{2}{2\cosh(x)} = \text{sech}(x) \), where \( \text{sech}(x) \) is the hyperbolic secant function.

Step 2: Choose a Comparison Function

To determine convergence, we need a function that bounds \( \text{sech}(x) \) and has a known integral behavior over \( \mathbb{R} \). A common choice is the exponential function. We know \( \text{sech}(x) \leq e^{-|x|} \) because for large \( |x| \), \( \cosh(x) \approx e^{|x|}/2 \), meaning \( \text{sech}(x) \approx 2e^{-|x|} \).

Step 3: Analyze Comparison Function

Consider the integral \( \int_{-\infty}^{\infty} e^{-|x|} \, dx \). This can be split into two parts: \( \int_{-\infty}^{0} e^{x} \, dx \) and \( \int_{0}^{\infty} e^{-x} \, dx \). Both integrals are convergent with values 1, hence \( \int_{-\infty}^{\infty} e^{-|x|} \, dx = 2 \). Since \( e^{-|x|} \) converges, the comparison test implies our original integral will also converge.

Step 4: Conclusion on Convergence

Since \( \text{sech}(x) \leq e^{-|x|} \) and \( \int_{-\infty}^{\infty} e^{-|x|} \, dx \) converges, by the comparison test, the original integral \( \int_{-\infty}^{\infty} \frac{1}{e^x+e^{-x}} \, dx \) is convergent.

Key Concepts

Integration Techniques

Understanding integrals is a key component of calculus. Integration is essentially the reverse process of differentiation. There are numerous techniques to solve integrals effectively.
Each technique provides a toolset for tackling different types of functions.
Common integration techniques include:

• Substitution: Used to simplify the integral by changing the variable.
• Integration by Parts: A technique derived from the product rule of differentiation.
• Partial Fractions: Useful for rational functions, allowing them to be expressed as a sum of simpler fractions.
• Trigonometric Substitution: Involves replacing variables with trigonometric functions to simplify the integrand.
• Hyperbolic Substitution: Similar to trigonometric substitution but uses hyperbolic functions, which is relevant to our exercise.

In the context of this problem, hyperbolic substitution is particularly useful. Here, the integrand is expressed in terms of hyperbolic secant, \( \text{sech}(x) = \frac{2}{e^x + e^{-x}} \). By recognizing hyperbolic forms, the integration process becomes clearer and more straightforward.

Comparison Test

The comparison test is a powerful tool in determining the convergence of integrals. It allows us to compare a complex integral with a simpler one whose convergence properties we already know.
This test is particularly effective when dealing with improper integrals, which extend to infinity or have discontinuities.
To apply the comparison test:

• Identify a known function that bounds the integrand either above or below.
• Ensure the bounding function has a known convergent integral over the same interval.
• If the bounding function's integral is convergent, so is the original if it lies below or equal to it; otherwise, both are divergent if it lies above.

In our problem, we use the function \( e^{-|x|} \) to bound \( \text{sech}(x) \). Since \( \text{sech}(x) \) is always less than or equal to \( e^{-|x|} \) and the integral of \( e^{-|x|} \) over all real numbers converges (with a value of 2), this confirms the convergence of the original integral \( \int_{-\infty}^{\infty} \text{sech}(x) \, dx \).
This method elegantly combines our ability to recognize simpler convergent integrals with more complex scenarios.

Hyperbolic Functions

Hyperbolic functions resemble trigonometric functions but are based on hyperbolas rather than circles. They often appear in calculus, especially in problems involving exponential functions due to their unique properties and identities.
Hyperbolic functions include \( \sinh(x), \cosh(x), \) and \( \text{sech}(x) \), among others.
Key properties of hyperbolic functions:

• Similar identities to trigonometric functions, such as \( \cosh^2(x) - \sinh^2(x) = 1 \).
• Expressed using exponential functions: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) and \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
• Hyperbolic secant \( \text{sech}(x) = \frac{1}{\cosh(x)} \) is notably useful in integration.

In the exercise, \( \text{sech}(x) \) simplifies the integrand, making it easier to assess convergence through its relationship with exponential functions. This transformation highlights the utility of hyperbolic functions in solving integrals efficiently.