Evaluate The Integral: ∫ 1 3 X + 12 D X \int \frac{1}{3x+12} \, Dx ∫ 3 X + 12 1 ​ D X

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Introduction

In this article, we will delve into the world of calculus and evaluate the integral of a rational function. The integral in question is $\int \frac{1}{3x+12} \, dx$. This type of integral is known as a logarithmic integral, and it can be evaluated using a variety of techniques. In this article, we will explore the method of substitution to evaluate this integral.

Background

Before we begin, let's review some background information on logarithmic integrals. A logarithmic integral is an integral of the form $\int \frac{1}{f(x)} \, dx$, where $f(x)$ is a function of $x$. These integrals are called logarithmic because they can be evaluated using the natural logarithm function. The natural logarithm function is defined as $\ln(x) = \int_1^x \frac{1}{t} \, dt$.

Method of Substitution

To evaluate the integral $\int \frac{1}{3x+12} \, dx$, we will use the method of substitution. This method involves substituting a new variable into the integral, which allows us to simplify the integral and make it easier to evaluate. In this case, we will substitute $u = 3x + 12$. This substitution will allow us to rewrite the integral in terms of $u$, which will make it easier to evaluate.

Substitution

Let's perform the substitution $u = 3x + 12$. This means that $du = 3dx$, or $dx = \frac{1}{3}du$. We can now substitute these expressions into the integral:

$$\int \frac{1}{3x+12} \, dx = \int \frac{1}{u} \cdot \frac{1}{3} \, du$$

Simplifying the Integral

Now that we have substituted $u = 3x + 12$ into the integral, we can simplify the integral. We can combine the fractions to get:

$$\int \frac{1}{u} \cdot \frac{1}{3} \, du = \frac{1}{3} \int \frac{1}{u} \, du$$

Evaluating the Integral

Now that we have simplified the integral, we can evaluate it. The integral of $\frac{1}{u}$ is the natural logarithm function, so we can write:

$$\frac{1}{3} \int \frac{1}{u} \, du = \frac{1}{3} \ln|u| + C$$

Substituting Back

Now that we have evaluated the integral, we can substitute back in for $u$. We know that $u = 3x + 12$, so we can write:

$$\frac{1}{3} \ln|u| + C = \frac{1}{3} \ln|3x + 12| + C$$

Conclusion

In this article, we evaluated the integral $\int \frac{1}{3x+12} \, dx$ using the method of substitution. We substituted $u = 3x + 12$ into the integral, which allowed us to simplify the integral and make it easier to evaluate. We then evaluated the integral and substituted back in for $u$ to get the final answer.

Final Answer

The final answer to the integral $\int \frac{1}{3x+12} \, dx$ is:

$$\frac{1}{3} \ln|3x + 12| + C$$

Applications

This type of integral has many applications in mathematics and science. For example, it can be used to model population growth, chemical reactions, and electrical circuits. It can also be used to solve problems in physics, engineering, and economics.

Future Work

In the future, we can explore other methods for evaluating logarithmic integrals. We can also investigate the properties of logarithmic integrals and how they can be used to solve problems in mathematics and science.

References

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Sullivan
• [3] "Calculus: Early Transcendentals" by James Stewart

Additional Resources

• [1] Khan Academy: Calculus
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Calculus

Conclusion

In conclusion, we have evaluated the integral $\int \frac{1}{3x+12} \, dx$ using the method of substitution. We have also explored the properties of logarithmic integrals and their applications in mathematics and science. We hope that this article has provided a useful resource for students and professionals who are interested in calculus and its applications.

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Introduction

In our previous article, we evaluated the integral $\int \frac{1}{3x+12} \, dx$ using the method of substitution. In this article, we will answer some common questions that students and professionals may have about this type of integral.

Q&A

Q: What is the method of substitution?

A: The method of substitution is a technique used to evaluate integrals by substituting a new variable into the integral. This allows us to simplify the integral and make it easier to evaluate.

Q: Why do we use the method of substitution?

A: We use the method of substitution to evaluate integrals that are difficult to evaluate directly. By substituting a new variable into the integral, we can simplify the integral and make it easier to evaluate.

Q: What is the difference between the method of substitution and the method of integration by parts?

A: The method of substitution and the method of integration by parts are two different techniques used to evaluate integrals. The method of substitution involves substituting a new variable into the integral, while the method of integration by parts involves differentiating and integrating a function.

Q: Can we use the method of substitution to evaluate any type of integral?

A: No, we cannot use the method of substitution to evaluate any type of integral. The method of substitution is only useful for evaluating integrals that can be simplified by substituting a new variable.

Q: What are some common mistakes to avoid when using the method of substitution?

A: Some common mistakes to avoid when using the method of substitution include:

• Not substituting the correct variable into the integral
• Not simplifying the integral correctly
• Not evaluating the integral correctly

Q: How do we know when to use the method of substitution?

A: We know when to use the method of substitution when we are faced with an integral that is difficult to evaluate directly. We can also use the method of substitution when we are given a hint or a suggestion to substitute a new variable into the integral.

Q: Can we use the method of substitution to evaluate integrals with multiple variables?

A: Yes, we can use the method of substitution to evaluate integrals with multiple variables. However, we need to be careful to substitute the correct variables into the integral and to simplify the integral correctly.

Q: What are some common applications of the method of substitution?

A: Some common applications of the method of substitution include:

• Evaluating integrals in physics and engineering
• Modeling population growth and chemical reactions
• Solving problems in economics and finance

Conclusion

In conclusion, we have answered some common questions about the method of substitution and its applications. We hope that this article has provided a useful resource for students and professionals who are interested in calculus and its applications.

Additional Resources

• [1] Khan Academy: Calculus
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Calculus

References

• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Sullivan
• [3] "Calculus: Early Transcendentals" by James Stewart

Future Work

In the future, we can explore other methods for evaluating integrals and their applications in mathematics and science. We can also investigate the properties of integrals and how they can be used to solve problems in physics, engineering, and economics.

Final Answer

The final answer to the integral $\int \frac{1}{3x+12} \, dx$ is:

$$\frac{1}{3} \ln|3x + 12| + C$$