Problem 49 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{\operatorname{sech} \sqrt{t} \tanh \sqrt{t} d t}{\sqrt{t}}$$

Step-by-Step Solution

Answer

The integral evaluates to \(-2 \operatorname{sech} \sqrt{t} + C\).

Step 1: Identify the Inner Function

Notice the presence of the terms \( \operatorname{sech} \sqrt{t} \) and \( \tanh \sqrt{t} \). This suggests that the argument of these functions \( \sqrt{t} \) should be used for substitution. Hence, let \( u = \sqrt{t} \), implying \( t = u^2 \) and \( dt = 2u \, du \).

Step 2: Substitute the Variables

Substitute \( u = \sqrt{t} \) into the integral. The expression becomes:\[ \int \frac{\operatorname{sech} u \tanh u \, 2u \, du}{u} = 2 \int \operatorname{sech} u \tanh u \, du \]The \( u \, du \) from \( dt \) cancels with the \( 1/u \).

Step 3: Evaluate the Simplified Integral

We now evaluate \( 2 \int \operatorname{sech} u \tanh u \, du \). The derivative of \( \operatorname{sech} u \) is \( -\operatorname{sech} u \tanh u \), which implies that the antiderivative is \(-\operatorname{sech} u \). Thus, we have: \[2 \left(-\int \operatorname{sech} u \tanh u \, du \right) = 2(-\operatorname{sech} u) \]

Step 4: Substitute Back to Original Variable

Replace \( u \) with \( \sqrt{t} \) to express the antiderivative in terms of the original variable: \[-2 \operatorname{sech} \sqrt{t} + C \]where \( C \) is the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions similar in form to trigonometric functions but with some distinct differences. They include functions like hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), and hyperbolic secant (sech), among others.

These functions are defined using the exponential function, often resulting in unique properties. For instance, the hyperbolic tangent, represented as \( \tanh(x) \), and the hyperbolic secant, written as \( \operatorname{sech}(x) \), are especially relevant when dealing with areas of calculus like differential equations and integrals.

• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)
• \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \)

These identities illustrate their close resemblance to trigonometric functions, providing a basis for solving complex mathematical problems.

Substitution Method

The substitution method is a powerful technique in calculus, often used to simplify integration problems. The fundamental idea is to substitute a part of the integrand with a new variable, making the integral easier to handle.

In the original exercise, we use substitution by setting \( u = \sqrt{t} \). This transformation changes the variables, simplifying the integral significantly.

• Substitute to simplify the integrals
• Transform complicated expressions into simple ones
• Apply derivative relationships

Once the substitution is complete, and the problem becomes manageable, reversing the substitution restores the expression to the original variables.

Antiderivatives

Finding antiderivatives, or indefinite integrals, involves determining a function whose derivative matches a given function. It is essentially the reverse process of differentiation.

The exercise involves evaluating a new integral after substitution. Recognizing that the derivative of \( \operatorname{sech}(u) \) is \( -\operatorname{sech}(u) \tanh(u) \), we can easily determine its antiderivative.

• Antiderivative reverses differentiation
• Solve integrals through recognizing patterns
• Often requires substitution to simplify

Once you find the antiderivative, you replace back any substitution variable to express the result in terms of the original variable, accompanied by a constant of integration \( C \).

Sech and Tanh Functions

The hyperbolic functions \( \operatorname{sech}(x) \) and \( \tanh(x) \) are essential when tackling integrals involving hyperbolic functions. The original exercise uses these functions, aiding in finding the solution.

Each function has unique properties and relations to its hyperbolic counterparts:

• \( \tanh(x) \) represents the hyperbolic tangent and is akin to trigonometric tangent
• \( \operatorname{sech}(x) \) is analogous to the reciprocal of hyperbolic cosine

Why are Sech and Tanh Important?

Understanding \( \operatorname{sech}(x) \) and \( \tanh(x) \) helps solve integrals and differential equations that may seem difficult initially. Like with trigonometric identities, knowing their relationships can simplify the evaluation of complex expressions.