Problem 60 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(51-60 .\) $$ \int_{0}^{\ln 10} 4 \sinh ^{2}\left(\frac{x}{2}\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2 \ln 10 - \frac{99}{10} \).

Step 1: Recall the formula for hyperbolic sine

The hyperbolic sine function is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). We are dealing with \( \sinh^2\left(\frac{x}{2}\right) \) in this integral.

Step 2: Use hyperbolic identity

To simplify the integral, use the identity \( \sinh^2(u) = \frac{1 - \cosh(2u)}{2} \). Therefore, \( \sinh^2\left(\frac{x}{2}\right) = \frac{1 - \cosh(x)}{2} \).

Step 3: Rewrite the integral

Substitute the identity into the integral to get: \[ \int_{0}^{\ln 10} 4 \left(\frac{1 - \cosh(x)}{2}\right) dx = \int_{0}^{\ln 10} 2 \left(1 - \cosh(x)\right) dx \].

Step 4: Split the integral

Separate the integral into two parts: \[ \int_{0}^{\ln 10} 2 dx - \int_{0}^{\ln 10} 2 \cosh(x) dx \].

Step 5: Integrate each term separately

For the first integral, \( \int_{0}^{\ln 10} 2 dx = 2x \big|_0^{\ln 10} = 2 \ln 10 \). For the second integral, use \( \int \cosh(x) dx = \sinh(x) \), giving us \( 2 \sinh(x) \big|_0^{\ln 10} \).

Step 6: Evaluate \( 2 \sinh(x) \)

Calculate \( 2 \sinh(x) \) at \( x = \ln 10 \) and \( x = 0 \): \( 2 \sinh(\ln 10) = 2 \left(\frac{e^{\ln 10} - e^{-\ln 10}}{2}\right) = 10 - \frac{1}{10} \) and \( 2 \sinh(0) = 0 \).

Step 7: Calculate the definite integral

Plug in the evaluated terms: \[ 2 \ln 10 - (10 - \frac{1}{10} - 0) = 2 \ln 10 - 10 + \frac{1}{10} \].

Step 8: Simplify the result

Combine the terms to get a single expression: \( 2 \ln 10 - \frac{99}{10} \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions play a crucial role in calculus, special functions, and mathematical modeling. They resemble trigonometric functions but have a distinct definition and properties. The hyperbolic sine, denoted as \( \sinh(x) \), is particularly important and is defined by the equation: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] These functions arise naturally in the study of certain physical phenomena, such as constructing catenary curves that describe the shape of a hanging cable or chain.

Another useful hyperbolic identity, relevant here, is \( \sinh^2(u) = \frac{1 - \cosh(2u)}{2} \). This helps simplify expressions involving hyperbolic functions, making them easier to integrate and manipulate.

Understanding hyperbolic functions gives students tools to solve real-world problems where exponential growth and decay are factors. By mastering these concepts, students can apply them in fields such as engineering, physics, and economic modeling.

Integration Techniques

In calculus, integration is a fundamental technique used to calculate areas under curves, among other things. Evaluating definite integrals, such as \( \int_{a}^{b} f(x) \, dx \), requires utilizing various integration techniques to find the antiderivative of a function. Here are some key techniques that are often used:

• Substitution: Allows for simplifying integrals by changing variables.
• Integration by Parts: Useful for products of functions.
• Trigonometric Identities: Simplify integrals by using trigonometric identities.
• Hyperbolic Identities: Similar to trigonometric identities but for hyperbolic functions, as shown in this problem.

In this exercise, we employed a hyperbolic identity to transform \( \sinh^2\left(\frac{x}{2}\right) \) into a form that was easier to integrate. This enabled us to break the integral down into simpler components: \[ \int_{0}^{\ln 10} 2 \, dx - \int_{0}^{\ln 10} 2 \cosh(x) \, dx \].

By working methodically, students can understand how various integration techniques provide powerful tools to tackle a wide range of problems.

Calculus Education

Calculus is a pivotal subject in mathematics education because it provides the foundation for understanding complex changes in mathematical models. It is instrumental in STEM (Science, Technology, Engineering, and Mathematics) education, and is crucial for fields like physics, engineering, and economics. Here are some teaching tips to improve calculus education:

• Visual Aids: Use graphs to illustrate concepts. Seeing functions and their integrals helps in visual understanding.
• Step-by-Step Examples: Work through problems step-by-step, as shown in this exercise, allowing students to follow the logical progression.
• Relate to Real-world Problems: Show how calculus solves real-world issues, making the subject more relatable and less abstract.
• Encourage Practice: Mastery only comes with practice. Encourage working on diverse problems.

Fostering a deep understanding of calculus not only aids academic success but also prepares students for future careers that rely heavily on mathematical reasoning and problem-solving skills.