Problem 47 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \operatorname{sech}^{2}\left(x-\frac{1}{2}\right) d x$$

Step-by-Step Solution

Answer

\( \tanh(x - \frac{1}{2}) + C \).

Step 1: Identify the Basic Integral Form

The integral \( \int \operatorname{sech}^2(u) \, du \) is a basic integral form that results in \( \tanh(u) + C \), where \( C \) is the integration constant. Our task is to transform the given integral into this form.

Step 2: Choose a Substitution

Let's set \( u = x - \frac{1}{2} \). This choice transforms the argument of the \( \operatorname{sech}^2 \) function into \( u \).

Step 3: Differentiate the Substitution

Differentiate both sides of the substitution \( u = x - \frac{1}{2} \) to obtain \( du = dx \). This means that our substitution simplifies the integral to a basic form.

Step 4: Rewrite the Integral in terms of u

We can now rewrite the original integral as \( \int \operatorname{sech}^2(u) \, du \), making it a straightforward application of a basic integral form.

Step 5: Integrate Using the Basic Form

Using the integral formula \( \int \operatorname{sech}^2(u) \, du = \tanh(u) + C \), we find that the integral evaluates to \( \tanh(u) + C \).

Step 6: Substituting Back to x

Recall our substitution \( u = x - \frac{1}{2} \); substitute back to get \( \tanh(x - \frac{1}{2}) + C \).

Step 7: Final Solution

The integral evaluates to \( \tanh(x - \frac{1}{2}) + C \), where \( C \) is the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but for a hyperbola. These include functions like hyperbolic sine, cosine, and tangent, denoted as \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \) respectively. One key formula related to hyperbolic functions is the definition of hyperbolic secant squared, \( \operatorname{sech}^2(x) = \frac{1}{\cosh^2(x)} \). This function is particularly important in calculus due to its straightforward integral form. The integral \( \int \operatorname{sech}^2(x) \, dx \) results in \( \tanh(x) + C \), where \( C \) is the integration constant.
Hyperbolic functions resemble trigonometric identities in many ways, however they uniquely handle certain types of equations involving exponential growth and decay. When dealing with integration, recognizing a hyperbolic function can save you time by allowing a quick substitution or transformation.
For example, in the current problem, identifying that the integral involves \( \operatorname{sech}^2(x) \) immediately suggests a direct application of the integral formula for \( \tanh(x) \). Understanding hyperbolic functions is therefore crucial for simplifying problems involving these functions.

Indefinite Integrals

Indefinite integrals represent a family of functions rather than a specific value. They express the antiderivative of a function and always include a constant of integration, \( C \). In simple terms, indefinite integrals help us find a broader function whose derivative is the given function.
When tackling indefinite integrals, the goal is often to simplify the expression to recognize a known integral form. Such simplifications often involve applying known antiderivative rules, such as those for polynomial, exponential, and especially trigonometric or hyperbolic functions.
In the exercise, we start with an integral expression \( \int \operatorname{sech}^2\left(x - \frac{1}{2}\right) dx \). By transforming it into the basic integral form, we utilize the definition that \( \int \operatorname{sech}^2(u) \, du \) results in \( \tanh(u) + C \). This showcases how recognizing integral forms simplifies evaluation and avoids unnecessary complications.

• Always remember: the constant \( C \) plays a crucial role since it accounts for all possible vertical shifts of the antiderivative graph.

Substitution Method

The substitution method in integration is a powerful technique used to simplify integrals. This method involves replacing a part of the integral with a new variable, making the integration process easier. It's similar to the reverse process of the chain rule used in differentiation.
To apply the substitution method, we choose a substitution (usually represented as \( u \)) that simplifies the integral. We then express \( dx \) in terms of \( du \), replace all \( x \) terms in the integral with \( u \) terms, and integrate with respect to \( u \).
In our problem, we set \( u = x - \frac{1}{2} \). This transformed the integral \( \int \operatorname{sech}^2\left(x - \frac{1}{2}\right) \, dx \) into \( \int \operatorname{sech}^2(u) \, du \), an integral form that is easier to evaluate directly. By using substitution, we reduced the complexity of the integration process.
Finally, after evaluating the integral in terms of \( u \), don’t forget to substitute back to the original variable \( x \). This brings the result back in line with the initial function context. The substitution method is especially useful when an integral doesn't initially resemble any basic form, providing a pathway to more straightforward calculation.