Problem 21 Find each indefinite integral. ... [FREE SOLUTION]

91影视

Brief Applied Calculus

Geoffrey C. Berresford, Andrew M. Rockett

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 21

Find each indefinite integral. $\int\left(e^{3 x}-\frac{3}{x}\right) d x$

Short Answer

Expert verified

The integral is $ \frac{1}{3} e^{3x} - 3 \ln|x| + C $.

Step by step solution

Integrate each term separately

Begin by splitting the integral into two separate integrals: $ \int e^{3x} \, dx - \int \frac{3}{x} \, dx $. This allows us to integrate each term individually.

Integrate the exponential function

The integral of $ e^{3x} $ can be found by using substitution. Let $ u = 3x $, thus $ du = 3 \, dx $ or $ dx = \frac{1}{3} \, du $. This transforms $ \int e^{3x} \, dx $ to $ \int \frac{1}{3} e^u \, du = \frac{1}{3} e^u + C_1 $. Substitute back $ u = 3x $ to get $ \frac{1}{3} e^{3x} + C_1 $.

Integrate the rational function

To integrate $ \int \frac{3}{x} \, dx $, recall that the integral of $ \frac{1}{x} $ is $ \ln|x| $. Therefore, $ \int \frac{3}{x} \, dx = 3 \ln|x| + C_2 $.

Combine the results and simplify

Combine the results from Steps 2 and 3: $ \frac{1}{3} e^{3x} - 3 \ln|x| + C $, where $ C $ is the combination of integration constants $ C_1 $ and $ C_2 $. This gives the complete indefinite integral.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function Integration

Integrating exponential functions can sometimes seem tricky, but once you understand a few simple steps, it becomes much easier. When dealing with integrals such as $ \int e^{3x} \, dx $, we use a method called substitution to simplify the process.

Here's how it works:

First, identify that the exponent in the function is a linear expression, like $ 3x $. We set $ u = 3x $, which simplifies our process.
Next, find the differential $ dx $ in terms of $ du $. For $ u = 3x $, differentiate to get $ du = 3 \cdot dx $, hence $ dx = \frac{1}{3} du $.
Substitute these into the integral: $ \int e^{3x} \, dx = \int e^u \frac{1}{3} du $.

The integral $ \int e^u \ du $ simplifies to $ e^u $. So, the integral becomes $ \frac{1}{3} e^u + C_1 $ where $ C_1 $ is a constant of integration. Finally, substitute $ u $ back to get $ \frac{1}{3} e^{3x} + C_1 $. This process allows you to find the indefinite integral of exponential functions effectively.

Rational Function Integration

Rational functions, like $ \frac{3}{x} $, often appear intimidating, but they aren't as complex as they seem. The key is to remember the natural logarithm function. Integrating $ \frac{3}{x} $ involves recalling a basic integral identity.

Here's a concise breakdown:

The integral $ \int \frac{1}{x} \, dx $ results in $ \ln|x| $. This is because the derivative of $ \ln|x| $ is $ \frac{1}{x} $.
Given $ \int \frac{3}{x} \, dx $, you can pull out the constant (3) outside the integral: $ 3 \int \frac{1}{x} \, dx $.

Once this is done, apply our integral identity: $ 3 \ln|x| + C_2 $, where $ C_2 $ is another constant of integration.

By focusing on core identities like $ \frac{1}{x} \Rightarrow \ln|x| $, rational integrations can be tackled with ease and precision.

Integration by Substitution

The process of integration by substitution is a vital skill in calculus, often used to simplify complex integrals. In our given problem, you may have noticed it was applied to the exponential component of the integral: $ \int e^{3x} \, dx $.

This method follows these basic steps:

Choose a substitution that simplifies the integral. Typically, this involves substituting a complicated expression with a simpler variable $ u $, like $ u = 3x $.
Differentiate this substitution to find $ du $, which helps in rewriting $ dx $ in terms of $ du $.
Replace the original variable and differential with $ u $ and $ du $ in the integral. This often turns the integral into a more straightforward problem.

After integrating with respect to $ u $, always substitute back the original variable to get a solution in terms of the initial variable, which in our exercise was $ x $.

Integration by substitution is akin to undoing the chain rule from differentiation, making it an essential and powerful technique in calculus.

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91影视

Find each indefinite integral. \(\int\left(e^{3 x}-\frac{3}{x}\right) d x\)

Short Answer

Step by step solution

Integrate each term separately

Integrate the exponential function

Integrate the rational function

Combine the results and simplify

Key Concepts

Exponential Function Integration

Rational Function Integration

Integration by Substitution

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