Problem 52 - Calculus of a Single Variable

Question

In Exercises 43–54, find the indefinite integral. $$ \int \operatorname{sech}^{3} x \tanh x d x $$

Step-by-Step Solution

Answer

The indefinite integral \( \int \operatorname{sech}^3 x \tanh x dx \) is equal to \(\tanh^3 x / 3 + C \)

Step 1: Setting up substitution

It can be seen that the derivative of \(\tanh x\) is \(\operatorname{sech}^2 x\). This suggests that a substitution of the form: let \(u = \tanh x\) and \(du = \operatorname{sech}^2 x\: dx\) will simplify the integral.

Step 2: Applying substitution and simplifying

Substitute \(u = \tanh x\) and \(du = \operatorname{sech}^2 x\: dx\) into the integral to get: \(\int u^2 du\)

Step 3: Integration

Now, integrate the resulting expression using the power rule of integration \(\int x^n dx = x^{n+1}/(n+1) + C\): \( \int u^2 du = u^3/3 + C \)

Step 4: Substitution back into original variable

Substitute \(u = \tanh x\) back into the equation: \( \int \operatorname{sech}^3 x \tanh x dx = \tanh^3 x / 3 + C \)

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus for simplifying the process of finding indefinite integrals. It often involves changing variables to make the integral more manageable and easier to solve. In our example, \(\int \operatorname{sech}^{3} x \tanh x \, dx\), you can notice that the derivative of \(\tanh x\) is \(\operatorname{sech}^2 x\). This insight helps us choose our substitution.

Here's why substitution works so well:

• By substituting \(u = \tanh x\), the integral \(\int \operatorname{sech}^{3} x \tanh x \, dx\) simplifies because the \(\operatorname{sech}^2 x\) component matches \(du\).
• This effectively transforms the integral into \(\int u^2 \, du\), a simpler form directly solvable using basic integration rules.

Substitution is particularly useful in integrals involving composition of functions where one part's derivative is present as another part, triggering a simpler format for integration.

Hyperbolic Functions

Hyperbolic functions, similar to their trigonometric counterparts, have unique properties that make them useful in calculus and integrating complex functions. The functions \(\sinh\), \(\cosh\), \(\tanh\), and \(\operatorname{sech}\) relate closely in structure to sine, cosine, tangent, and secant, respectively.

For this exercise, understanding the derivatives and relationships between these functions was crucial:

• Derivative of \(\tanh x\) is \(\operatorname{sech}^2 x\), used in our substitution.
• Hyperbolic identities allow transformations similar to trigonometric identities, making complicated integrals manageable.

Hyperbolic functions appear in many real-world applications, such as describing the shape of cables or arches. Hence, mastering these functions extends beyond calculus class into practical, physical problem-solving.

Integration Techniques

Mastering various integration techniques is essential for solving a broad range of integral problems. Each technique has its optimal use case.

In this exercise, the power rule is employed after the substitution transforms the given integral into a basic polynomial form \(\int u^2 \, du\). The power rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), a straightforward formula to apply with monomials.

Other integration techniques include:

• Integration by parts for products of functions.
• Partial fraction decomposition, useful for rational functions.
• Trigonometric substitutions helpful when integrating functions involving radicals.

Choosing the right technique depends largely on recognizing patterns within the integral. Practicing a diverse set of problems equips students to select and apply the right method efficiently.