Problem 45 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-50\) . $$ \int \tanh \frac{x}{7} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 7 \ln|\cosh(\frac{x}{7})| + C \).

Step 1: Recall the Properties of Hyperbolic Functions

The hyperbolic tangent function, \( \tanh(x) \), is defined as \( \frac{\sinh(x)}{\cosh(x)} \). The derivative of \( \tanh(x) \) is \( \text{sech}^2(x) \). We will need these definitions for solving the integral.

Step 2: Make a Substitution

To simplify the integral \( \int \tanh\left(\frac{x}{7}\right) dx \), use the substitution \( u = \frac{x}{7} \), hence \( du = \frac{1}{7} dx \) or \( dx = 7 du \). Substitute these into the integral to transform it.

Step 3: Rewrite the Integral in Terms of u

With the substitution from Step 2, the integral becomes \( \int \tanh(u) \cdot 7 \, du \), which simplifies to \( 7 \int \tanh(u) \, du \).

Step 4: Integrate \( \tanh(u) \)

The integral \( \int \tanh(u) \, du \) is known to equal \( \ln|\cosh(u)| + C \), where \( C \) is the integration constant. Use this result to continue.

Step 5: Back-Substitute to x

Since \( u = \frac{x}{7} \), replace \( u \) in the integrated function to return to terms of \( x \). This gives us \( 7 \ln|\cosh\left(\frac{x}{7}\right)| + C \).

Step 6: Final Answer

Combining all the steps above, the integral \( \int \tanh\left(\frac{x}{7}\right) \, dx \) is \( 7 \ln|\cosh\left(\frac{x}{7}\right)| + C \), where \( C \) is the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, similar to trigonometric functions, are derived from the hyperbola. They include functions like hyperbolic sine, hyperbolic cosine, and hyperbolic tangent, which are often denoted as \( \sinh(x) \), \( \cosh(x) \), and \( \tanh(x) \) respectively.
These functions are essential in both pure and applied mathematics, particularly in calculus and engineering applications.
The hyperbolic tangent function, \( \tanh(x) \), is defined as:

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

It shares similar properties with the regular tangent function, but is defined using exponential functions:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

Understanding these definitions helps greatly in performing calculus operations such as differentiation and integration involving hyperbolic functions.

Substitution Method

The substitution method is a classic technique used to simplify challenging integrals. Its main idea is to change variables to make integration more manageable. In the given exercise, a substitution is made by setting \( u = \frac{x}{7} \).
This transforms the differential \( dx \) into a new format derived from \( du \):

• \( du = \frac{1}{7}dx \)
• which implies \( dx = 7du \)

Substituting \( u \) into a function changes the integral's structure from something difficult to integrate to something more familiar.
In our example, it changes from \( \int \tanh\left(\frac{x}{7}\right) dx \) into \( 7 \int \tanh(u) du \), a form that is often easier to solve. This method, sometimes referred to as "u-substitution," is widely applied to various problems in integral calculus.

Integral of Hyperbolic Tangent

Integrating hyperbolic functions can initially seem daunting, but they often follow clear patterns once the properties are understood.
For the integral \( \int \tanh(u) du \), the antiderivative is knowable:

• \( \int \tanh(u) du = \ln|\cosh(u)| + C \)

Where \( C \) stands as the constant of integration. This result is derived from the relationship between the derivative of the hyperbolic tangent function and the secant hyperbolic function: \( \text{sech}^2(u) \).
Remember, integrating hyperbolic functions often involves understanding their derivatives and recognizing the corresponding antiderivative forms. This helps in applications involving exponential growth, relativistic physics, and hyperbolic geometry.

Integration Techniques

Integration techniques are critical tools in calculus, allowing us to solve a wide variety of integrals.
Several core methods exist, one of which includes substitution, as we applied earlier, aiding in simplifying complex integrals. Other methods complement substitution, such as:

• Integration by parts
• Partial fraction decomposition
• Trigonometric identities and trigonometric substitution

Beyond these, recognizing patterns is crucial. Knowing the fundamental antiderivatives, such as those of hyperbolic functions, expands your toolkit.
This transforms integration from a challenging task into a more methodical and manageable process, allowing you to tackle more advanced problems confidently. Each technique has its specific uses and strengths, helping manage different algebraic and transcendental forms in calculus.