Problem 11 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=5 e^{x}-3 \cosh x$$

Step-by-Step Solution

Answer

The most general antiderivative is \( 5e^x - 3\sinh x + C \).

Step 1: Introduction to the Problem

We are tasked with finding the most general antiderivative of the function \( f(x) = 5e^x - 3 ext{cosh} \, x \) and then verifying our result through differentiation.

Step 1: Consider the Components Separately

The function consists of two terms: \( 5e^x \) and \(-3\cosh x\). We'll find the antiderivative of each term separately.

Step 2: Find the Antiderivative of \( 5e^x \)

The antiderivative of \( e^x \) is itself \( e^x \). Thus, the antiderivative of \( 5e^x \) is \( 5e^x \).

Step 3: Find the Antiderivative of \(-3\cosh x\)

The hyperbolic cosine function \( \cosh x \) has an antiderivative of \( \sinh x \). Therefore, the antiderivative of \(-3\cosh x\) is \(-3\sinh x \).

Step 4: Combine the Antiderivatives

Adding the results from Steps 2 and 3, we have the antiderivative: \( 5e^x - 3\sinh x + C \) where \( C \) is the constant of integration.

Step 5: Verify by Differentiation

Differentiate the antiderivative \( 5e^x - 3\sinh x + C \). - The derivative of \( 5e^x \) is \( 5e^x \). - The derivative of \( -3\sinh x \) is \(-3\cosh x \). - The derivative of \( C \) is 0. Thus, the differentiation confirms \( f(x) = 5e^x - 3\cosh x \).

Key Concepts

Understanding Integration Techniques

When working with integration techniques, it's all about finding the antiderivative of a function, which essentially means finding the original function whose derivative gives you the function you started with.
In the exercise, we are dealing with two separate terms: \(5e^x\) and \(-3\cosh x\). Finding their antiderivatives involves knowing well-known basic integration rules.Here are key points:

• For the term \(5e^x\), remember that the exponential function \(e^x\) is unique. Its derivative and antiderivative are both \(e^x\), allowing us to directly write the antiderivative of \(5e^x\) as \(5e^x\).
• For the term involving \(-3\cosh x\), understanding hyperbolic functions is crucial. Knowing that the derivative of \(\sinh x\) is \(\cosh x\) allows us to determine that the antiderivative of \(-3\cosh x\) is \(-3\sinh x\).
• The constant of integration, \(C\), is added to represent the family of all possible antiderivatives, as the derivative of any constant is zero.

Mastering these integration techniques is vital for solving many calculus problems, particularly when functions are composed of well-known derivatives and antiderivatives.

Exploring Hyperbolic Functions

Hyperbolic functions, similar to trigonometric functions, are important in calculus, but they correspond to a different kind of geometry, namely hyperbolic geometry.
The functions we often encounter are \(\cosh x\), \(\sinh x\), and others like \(\tanh x\).For the exercise:

• The function \(\cosh x\), or hyperbolic cosine, is used, and its definition is based on exponential functions: \(\cosh x = \frac{e^x + e^{-x}}{2}\).
• Similarly, \(\sinh x\), or hyperbolic sine, is \(\sinh x = \frac{e^x - e^{-x}}{2}\). These relationships make it simpler to find derivatives and antiderivatives.
• Knowing that \(\frac{d}{dx}(\sinh x) = \cosh x\) and vice versa allows you to switch between these derivatives and antiderivatives with ease, crucial for solving complex integration problems.

Understanding the behavior and properties of hyperbolic functions helps not only in pure mathematics but also in fields like physics and engineering where they model natural phenomena.

Differentiation Verification

Differentiation verification is a powerful tool that ensures the accuracy of your antiderivative by differentiating it and checking if you return to the original function.
This process solidifies your solution.In practice:

• Start by taking your found antiderivative, in this case, \(5e^x - 3\sinh x + C\), and differentiating it. This involves applying derivative rules you are already familiar with.
• Differentiate \(5e^x\) and \(-3\sinh x\). Recall, the derivative of \(e^x\) is \(e^x\) and the derivative of \(-\sinh x\) is \(-\cosh x\). Therefore, you naturally retrieve \(5e^x - 3\cosh x\).
• The inclusion of \(C\) ensures you're considering the entire family of functions, though its derivative is zero and doesn't affect the expression.

This differentiation verification step is essential to confirm the correctness of your integration, ensuring no steps were missed and that the antiderivative genuinely represents the original function's family.