Problem 5

Question

Evaluate the indefinite integral by making the given substitution. $$ \int 5 \cos (3 x) d x, \text { with } u=3 x $$

Step-by-Step Solution

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Answer

The integral evaluates to $ \frac{5}{3} \sin(3x) + C $.

1Step 1: Identify the substitution

We are given the substitution $ u = 3x $. This means we will replace $ 3x $ in the integral with $ u $.

2Step 2: Differentiate the substitution

Find the differential $ du $ by differentiating $ u = 3x $ with respect to $ x $, which gives $ du = 3dx $.

3Step 3: Express dx in terms of du

From $ du = 3 dx $, solve for $ dx $ to get $ dx = \frac{du}{3} $.

4Step 4: Substitute in the integral

Substitute $ u = 3x $ and $ dx = \frac{du}{3} $ into the integral $ \int 5 \cos(3x) \, dx $, changing it to $ \int 5 \cos(u) \frac{du}{3} $.

5Step 5: Simplify the integral

Simplify by factoring constants out of the integral: $ \int \frac{5}{3} \cos(u) \, du $. This becomes $ \frac{5}{3} \int \cos(u) \, du $.

6Step 6: Integrate

The integral of $ \cos(u) $ is $ \sin(u) $, so the solution of $ \int \cos(u) \, du $ is $ \sin(u) + C $. Thus, $ \frac{5}{3} \int \cos(u) \mathrm{d}u $ becomes $ \frac{5}{3} \sin(u) + C $.

7Step 7: Substitute back

Replace $ u $ with $ 3x $ to return to the original variable. This gives $ \frac{5}{3} \sin(3x) + C $.

Key Concepts

Substitution MethodIntegration TechniquesTrigonometric Integrals

Substitution Method

The substitution method is a powerful tool in calculus for evaluating integrals, especially when dealing with composite functions. It involves substituting part of the integrand with a new variable to simplify the integral.
This method is analogous to the chain rule for differentiation, but instead applied to integration.
In this problem, the substitution given is $ u = 3x $. This substitution aims to simplify the trigonometric expression $ \cos(3x) $ into a simpler form, $ \cos(u) $.

First, identify the part of the function that makes substitution beneficial – typically, this is the inside function of a composite.
Differentiate the substitution expression to find $ du = 3dx $ so that you can replace $ dx $ with $ \frac{du}{3} $.
Substitute both $ u $ and $ \frac{du}{3} $ back into the integral, which transforms the original complex expression into a more manageable one.

The goal is to transform the integral so that you'll end up integrating a simpler function. After solving, make sure to substitute back the original variable to complete the process.

Integration Techniques

Integration techniques are essential in calculus for finding antiderivatives of various functions. There are several techniques available, each suited for different kinds of problems.
The substitution method, as used in this task, is just one of these techniques, tailored particularly to functions where a piece can be isolated and replaced with a single variable.
One crucial aspect of integration is factoring constants. Here, we took $ \frac{5}{3} $ outside of the integral sign.

Taking constants outside the integral simplifies the expression, helping focus on the integration of the core variable-dependent part.
We simplify $ \int \, \frac{5}{3} \cos(u) \, du $ to $ \frac{5}{3} \int \cos(u) \, du $, making it easier to apply basic integration formulas.

At the end of the integration, always remember to add the constant of integration $ C $, since it's an indefinite integral. This represents the general solution of an antiderivative which could vary by any constant.

Trigonometric Integrals

Trigonometric integrals appear frequently in calculus problems and usually involve integrating a function with trigonometric expressions like sine or cosine.
For the problem $ \int 5 \cos(3x) \, dx $, using the substitution $ u = 3x $, simplifies the complex trigonometric expressions.
A few key steps in addressing trigonometric integrals:

Recognize the form of the trigonometric function; e.g., $ \cos(u) $ is straightforward under substitution.
Know the basic antiderivatives of trigonometric functions. The integral of $ \cos(u) $ is $ \sin(u) + C $.
After integration, revert all substitutions to express the result in the original variable, in this case, replacing $ u $ with $ 3x $.

This process ultimately returns the solution to the original trigonometric integral problem: $ \frac{5}{3} \sin(3x) + C $. Emphasizing understanding and memorizing basic trigonometric integrals can significantly ease the solving process for more complex equations.

Problem 5

Other exercises in this chapter

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