3 Integrations That My Teacher Gave That I Have No Idea How To Solve.

3 Integrations that Left Me Stumped: A Guide to Solving Complex Integrals

As a student, we often encounter complex integrals that leave us scratching our heads. In this article, we will explore three integrations that my teacher assigned, and I had no idea how to solve. We will break down each problem step by step, and provide a clear explanation of the solution. By the end of this article, you will have a better understanding of how to tackle complex integrals and become more confident in your problem-solving skills.

Integration 1: A Substitution Gone Wrong

Problem Statement

$$\int \frac{x + 1}{x(1 + xe^x)} \, dx$$

My Attempts

I attempted a basic substitution, but other than that, I'm stumped.

$$1+xe^x =t$$

$$\frac{d}{dx}(1 + x e^x) = \frac{dt}{dx} = e^x(x + 1)$$

Solution

To solve this integral, we need to use a more sophisticated substitution. Let's start by rewriting the integral:

$$\int \frac{x + 1}{x(1 + xe^x)} \, dx = \int \frac{1}{1 + xe^x} \, dx$$

Now, let's use the substitution:

$$u = 1 + xe^x$$

$$du = (x + 1)e^x \, dx$$

We can rewrite the integral in terms of u:

$$\int \frac{1}{u} \, du$$

This is a standard integral, and the solution is:

$$\ln|u| + C$$

Now, we need to substitute back in terms of x:

$$\ln|1 + xe^x| + C$$

Conclusion

In this section, we solved a complex integral using a sophisticated substitution. We started by rewriting the integral, and then used the substitution to simplify the expression. By following these steps, we were able to find the solution to the integral.

Integration 2: A Trigonometric Integral

Problem Statement

$$\int \frac{1}{\sqrt{1 - \sin^2x}} \, dx$$

Solution

To solve this integral, we need to use trigonometric identities. Let's start by rewriting the integral:

$$\int \frac{1}{\sqrt{1 - \sin^2x}} \, dx = \int \frac{1}{\sqrt{\cos^2x}} \, dx$$

Now, we can use the identity:

$$\cos^2x = 1 - \sin^2x$$

We can rewrite the integral as:

$$\int \frac{1}{\sqrt{\cos^2x}} \, dx = \int \frac{1}{|\cos x|} \, dx$$

This is a standard integral, and the solution is:

$$\ln|\sec x + \tan x| + C$$

Conclusion

In this section, we solved a trigonometric integral using trigonometric identities. We started by rewriting the integral, and then used the identity to simplify the expression. By following these steps, we were able to find the solution to the integral.

Integration 3: A Logarithmic Integral

Problem Statement

$$\int \frac{1}{x \ln x} \, dx$$

Solution

To solve this integral, we need to use logarithmic properties. Let's start by rewriting the integral:

$$\int \frac{1}{x \ln x} \, dx = \int \frac{1}{x} \cdot \frac{1}{\ln x} \, dx$$

Now, we can use the property:

$$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

We can rewrite the integral as:

$$\int \frac{1}{x} \cdot \frac{1}{\ln x} \, dx = \int \frac{1}{\ln x} \, d(\ln x)$$

This is a standard integral, and the solution is:

$$\ln|\ln x| + C$$

Conclusion

In this section, we solved a logarithmic integral using logarithmic properties. We started by rewriting the integral, and then used the property to simplify the expression. By following these steps, we were able to find the solution to the integral.

In this article, we explored three integrations that my teacher assigned, and I had no idea how to solve. We broke down each problem step by step, and provided a clear explanation of the solution. By following these steps, you will have a better understanding of how to tackle complex integrals and become more confident in your problem-solving skills. Remember to always start by rewriting the integral, and then use the appropriate substitution or identity to simplify the expression. With practice and patience, you will become proficient in solving complex integrals and excel in your math studies.
3 Integrations that Left Me Stumped: A Guide to Solving Complex Integrals - Q&A

As a student, we often encounter complex integrals that leave us scratching our heads. In our previous article, we explored three integrations that my teacher assigned, and I had no idea how to solve. We broke down each problem step by step, and provided a clear explanation of the solution. In this article, we will answer some of the most frequently asked questions about complex integrals and provide additional tips and resources for solving them.

Q: What is the most common mistake students make when solving complex integrals?

A: The most common mistake students make when solving complex integrals is not rewriting the integral in a simpler form. This can make it difficult to identify the correct substitution or identity to use.

Q: How do I know which substitution to use when solving a complex integral?

A: When solving a complex integral, you need to identify the correct substitution to use. This can be done by rewriting the integral in a simpler form and looking for patterns or relationships between the variables.

Q: What is the difference between a substitution and an identity?

A: A substitution is a change of variables that simplifies the integral, while an identity is a mathematical relationship between two or more variables that can be used to simplify the integral.

Q: How do I know when to use a trigonometric identity when solving a complex integral?

A: When solving a complex integral, you need to look for trigonometric relationships between the variables. If you see a trigonometric function, such as sine or cosine, you may be able to use a trigonometric identity to simplify the integral.

Q: What is the most important thing to remember when solving complex integrals?

A: The most important thing to remember when solving complex integrals is to be patient and persistent. Complex integrals can be challenging to solve, but with practice and patience, you will become proficient in solving them.

• Practice, practice, practice: The best way to become proficient in solving complex integrals is to practice, practice, practice.
• Use online resources: There are many online resources available that can help you solve complex integrals, including video tutorials, practice problems, and online calculators.
• Seek help from a teacher or tutor: If you are struggling to solve a complex integral, don't be afraid to seek help from a teacher or tutor.
• Read and understand the problem: Before starting to solve a complex integral, make sure you understand the problem and what is being asked.

In this article, we answered some of the most frequently asked questions about complex integrals and provided additional tips and resources for solving them. By following these tips and using the resources available, you will become proficient in solving complex integrals and excel in your math studies.

Here are some common integrals and their solutions:

• ∫(1/x) dx: This is a standard integral that has a solution of ln|x| + C.
• ∫(1/(1 + x^2)) dx: This is a standard integral that has a solution ofctan(x) + C.
• ∫(e^x) dx: This is a standard integral that has a solution of e^x + C.

In conclusion, complex integrals can be challenging to solve, but with practice and patience, you will become proficient in solving them. Remember to always start by rewriting the integral, and then use the appropriate substitution or identity to simplify the expression. With the tips and resources provided in this article, you will be well on your way to becoming a master of complex integrals.