Problem 40 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x $$

Step-by-Step Solution

Answer

The integral is \( \ln|\cosh(x)| + C \).

Step 1: Simplify the Expression

First, recognize that the given expression, \( \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \), can be simplified using known hyperbolic functions. This expression is equivalent to \( \tanh(x) \), the hyperbolic tangent function. Therefore, the integral becomes \( \int \tanh(x) \, dx \).

Step 2: Identify the Integral Formula

The integral of \( \tanh(x) \) is a standard result which equals \( \ln|\cosh(x)| + C \), where \( C \) is the arbitrary constant of integration.

Step 3: Solve the Integral

Therefore, by directly using the integral formula, the solution to the integral \( \int \tanh(x) \, dx \) is \( \ln|\cosh(x)| + C \).

Step 4: Differentiate to Verify the Answer

To verify, differentiate \( \ln|\cosh(x)| + C \) with respect to \( x \). The derivative is \( \frac{d}{dx} \ln|\cosh(x)| = \frac{\sinh(x)}{\cosh(x)} \), which simplifies to \( \tanh(x) \). Thus, differentiating returns the original integrand \( \tanh(x) \), confirming that the result is correct.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but for a hyperbola, rather than a circle. Similar to sine, cosine, and tangent in trigonometry, hyperbolic functions also have important counterparts: sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent). These functions are defined using exponential functions, such as:

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

The expression in our exercise, \( \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \), simplifies directly to \( \tanh(x) \). Understanding this simplification helps in recognizing patterns and solving integrals more effectively.

Integration by Recognition

Integration by recognition involves identifying standard integral forms within an expression so you can apply known results directly. This method is efficient and often used when you have a library of standard forms memorized. For instance, the integral \( \int \tanh(x) \, dx \) is known to be equivalent to \( \ln|\cosh(x)| + C \), where \( C \) is the constant of integration.
By recognizing that our given problem simplifies to the integral of \( \tanh(x) \), we apply this formula directly without performing intricate integration steps. This technique is particularly useful in exams or problem-solving situations where time is limited. Learning and practicing with common function forms can greatly speed up the integration process.

Verification by Differentiation

Verification by differentiation is the process of confirming the correctness of an integral solution by differentiating it back to see if you retrieve the original integrand. It is a crucial part of solving integrals, akin to checking your arithmetic in simpler math problems. When we differentiate the result of \( \int \tanh(x) \, dx \), which is \( \ln|\cosh(x)| + C \), we calculate:

• \( \frac{d}{dx} [\ln|\cosh(x)| + C] = \frac{\sinh(x)}{\cosh(x)} \)
• This simplifies to \( \tanh(x) \), confirming the result.

This verification step reassures us of the reliability of our solution and highlights the beautiful symmetry between integration and differentiation.