Problem 76 - Thomas Calculus

Question

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int 3 \sinh \left(\frac{x}{2}+\ln 5\right) d x $$

Step-by-Step Solution

Answer

The integral is \( 15 e^{\frac{x}{2}} - \frac{3}{5} e^{-\frac{x}{2}} + C \).

Step 1: Simplify the Hyperbolic Sine Function

The given integral is \( \int 3 \sinh \left(\frac{x}{2} + \ln 5\right) dx \). Recognize that the hyperbolic sine function, \( \sinh(u) \), is defined as \( \sinh(u) = \frac{e^u - e^{-u}}{2} \). So, the expression \( \sinh \left( \frac{x}{2} + \ln 5 \right) \) becomes \( \frac{1}{2} \left( e^{\frac{x}{2} + \ln 5} - e^{-\left(\frac{x}{2} + \ln 5\right)} \right) \).

Step 2: Rewrite Exponents

Use the properties of exponents to simplify the terms. The expression \( e^{\frac{x}{2} + \ln 5} \) becomes \( 5 e^{\frac{x}{2}} \) by recognizing it as \( e^{\frac{x}{2}} \cdot e^{\ln 5} \). Similarly, \( e^{-\left(\frac{x}{2} + \ln 5\right)} \) simplifies to \( \frac{e^{-\frac{x}{2}}}{5} \).

Step 3: Substitute into the Integral

Substitute the simplified form of the hyperbolic sine function into the integral:\[ \int 3 \left( \frac{1}{2} \left( 5 e^{\frac{x}{2}} - \frac{e^{-\frac{x}{2}}}{5} \right) \right) dx \] which simplifies to:\[ \int \left( \frac{15}{2} e^{\frac{x}{2}} - \frac{3}{10} e^{-\frac{x}{2}} \right) dx \].

Step 4: Integrate Each Term Separately

Integrate each term separately:- The integral of \( \frac{15}{2} e^{\frac{x}{2}} \) with respect to \( x \) is \( 15 e^{\frac{x}{2}} \) since the derivative of \( e^{\frac{x}{2}} \) with respect to \( x \) is \( \frac{1}{2} e^{\frac{x}{2}} \), multiplying by 2 gives \( 15 e^{\frac{x}{2}} \).- The integral of \( -\frac{3}{10} e^{-\frac{x}{2}} \) with respect to \( x \) is \( -\frac{3}{5} e^{-\frac{x}{2}} \) since the derivative of \( e^{-\frac{x}{2}} \) is \( -\frac{1}{2} e^{-\frac{x}{2}} \), thus multiplying by -2 gives \( -\frac{3}{5} e^{-\frac{x}{2}} \).

Step 5: Combine the Results and Add the Constant of Integration

Combine the results from Step 4 and add the constant of integration \( C \):\[ \int 3 \sinh \left(\frac{x}{2} + \ln 5\right) dx = 15 e^{\frac{x}{2}} - \frac{3}{5} e^{-\frac{x}{2}} + C \].

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They include functions like hyperbolic sine \(\sinh(x)\), hyperbolic cosine \(\cosh(x)\), and others. \(\sinh(x)\) is defined as:

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

These functions are useful in many areas of mathematics and physics, appearing in calculations for hanging cables, surfaces of revolution, and more. Understanding how hyperbolic functions are expressed in terms of exponential functions helps simplify integration. For example, when tackling the integral of \(\sinh(\frac{x}{2} + \ln 5)\), recognizing its relationship to exponential functions allows for breaking down complex expressions. Hyperbolic functions are vital tools, especially when techniques of solving integrals are involved.

Exponential Functions

Exponential functions have the form \(f(x) = a^x\), where \(a\) is a constant. A crucial property of exponential functions is their natural growth pattern, where they change by a percentage rate over time. The base of the natural logarithm, \(e\), is the most common base in calculus. These functions are integral to defining hyperbolic functions:

• \(e^x\) and \(e^{-x}\) form the building blocks of \(\sinh(x)\) and \(\cosh(x)\).

For integration purposes, understanding exponential rules, such as \(e^{a+b} = e^a \cdot e^b\), is essential. In simplifying integrals involving hyperbolic functions, exponential properties are frequently employed. In the original exercise, simplifying \(e^{\frac{x}{2} + \ln 5}\) by breaking it into \(5 e^{\frac{x}{2}}\) was a critical step achieved with exponential rules. Recognizing these connections helps in effectively managing complex calculus problems.

Integration Techniques

Integration techniques are methods for calculating integrals, the antiderivatives of functions. Several strategies exist:

• Substitution: Used for simplifying integrals by substituting variables to make them more straightforward.
• Integration by Parts: A technique used primarily for products of functions.
• Partial Fractions: Breaking complex fractions into simpler pieces that can be integrated individually.

For the integral \(\int 3 \sinh(\frac{x}{2} + \ln 5)\, dx\), breaking it down using the definition of \(\sinh\) as exponential functions simplified the process. Each term can be integrated independently once in exponential form. Integrating exponential functions directly, using their derivative properties, showcases the power of recognizing applicable techniques, making problems easier to solve.

Definite and Indefinite Integrals

Integrals can either be definite or indefinite.

• Indefinite integrals represent families of functions and include a constant of integration \(C\) to account for any constant shift.
• Definite integrals calculate a number, providing the area under a curve between two limits. They don’t include \(C\).

In solving integrals such as \(\int 3 \sinh(\frac{x}{2} + \ln 5)\, dx\), we perform an indefinite integral because we integrate the function without specific limits. The solution \(15 e^{\frac{x}{2}} - \frac{3}{5} e^{-\frac{x}{2}} + C\) reflects this, highlighting the constant term added. Understanding when to apply and calculate definite versus indefinite integrals is a foundational concept in calculus, crucial for solving various mathematical and real-world problems.