Use The Formula $\int U E^{a U} \, D U = \frac{1}{a^2}(a U - 1) E^{a U} + C$ To Find $\int X E^{5 X} \, D X$. Fill In The Box $\square$ With Your Answer And Add $+ C$.

Introduction

In calculus, integrals of exponential functions are a common and essential topic. The formula $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$ is a powerful tool for solving these types of integrals. In this article, we will use this formula to find the integral of $x e^{5 x}$.

The Formula

The given formula is $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$. This formula is a general solution for integrals of the form $\int u e^{a u} \, d u$. To use this formula, we need to identify the values of $u$ and $a$ in the given integral.

Identifying the Values of $u$ and $a$

In the given integral $\int x e^{5 x} \, d x$, we can identify $u = x$ and $a = 5$. Now that we have identified the values of $u$ and $a$, we can substitute them into the formula.

Substituting the Values into the Formula

Substituting $u = x$ and $a = 5$ into the formula, we get:

$\int x e^{5 x} \, d x = \frac{1}{5^2}(5 x - 1) e^{5 x} + C$

Simplifying the Expression

Simplifying the expression, we get:

$\int x e^{5 x} \, d x = \frac{1}{25}(5 x - 1) e^{5 x} + C$

The Final Answer

Therefore, the final answer is:

$\boxed{\frac{1}{25}(5 x - 1) e^{5 x} + C}$

Conclusion

In this article, we used the formula $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$ to find the integral of $x e^{5 x}$. We identified the values of $u$ and $a$, substituted them into the formula, and simplified the expression to get the final answer.

Additional Examples

Here are a few additional examples of using the formula to solve integrals of exponential functions:

• $\int x e^{2 x} \, d x = \frac{1}{2^2}(2 x - 1) e^{2 x} + C$
• $\int x e^{3 x} \, d x = \frac{1}{3^2}(3 x - 1) e^{3 x} + C$
• $\int x e^{4 x} \, d x = \frac{1}{4^2}(4 x - 1) e^{4 x} + C$

Tips and Tricks

Here are a few tips and tricks for using the formula to solve integrals of exponential functions:

• Make sure to identify the values of $u$ and $a$ correctly.
• Substitute the values into the formula carefully.
• Simplify the expression to get the final answer.

Common Mistakes

Here are a few common mistakes to avoid when using the formula to solve integrals of exponential functions:

• Not identifying the values of $u$ and $a$ correctly.
• Not substituting the values into the formula carefully.
• Not simplifying the expression to get the final answer.

Conclusion

Introduction

In our previous article, we used the formula $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$ to find the integral of $x e^{5 x}$. In this article, we will answer some frequently asked questions about using the formula to solve integrals of exponential functions.

Q: What is the formula for solving integrals of exponential functions?

A: The formula for solving integrals of exponential functions is $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$.

Q: How do I identify the values of $u$ and $a$ in the given integral?

A: To identify the values of $u$ and $a$, you need to look at the given integral and identify the function that is being integrated. In the case of $\int x e^{5 x} \, d x$, $u = x$ and $a = 5$.

Q: What if I have a different integral, such as $\int x e^{2 x} \, d x$? How do I use the formula?

A: To use the formula for $\int x e^{2 x} \, d x$, you need to identify the values of $u$ and $a$. In this case, $u = x$ and $a = 2$. Then, you can substitute these values into the formula and simplify the expression to get the final answer.

Q: What if I make a mistake when identifying the values of $u$ and $a$? How do I correct it?

A: If you make a mistake when identifying the values of $u$ and $a$, you need to go back and re-examine the given integral. Make sure to identify the function that is being integrated and the constant that is being multiplied by the function. Once you have correctly identified the values of $u$ and $a$, you can substitute them into the formula and simplify the expression to get the final answer.

Q: Can I use the formula to solve integrals of other types of functions, such as trigonometric functions?

A: No, the formula $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$ is only for solving integrals of exponential functions. If you have an integral of a trigonometric function, you will need to use a different formula or technique to solve it.

Q: How do I know if I have used the formula correctly?

A: To know if you have used the formula correctly, you need to check your work carefully. Make sure to identify the values of $u$ and $a$ correctly, substitute them into the formula, and simplify the expression to get the final answer. If you have made a mistake, you will need to go back and re-examine your work.

Q: What if I get a different answer than the one in the textbook or online resource? How do I know which answer is correct?

A: If you get a different answer than the one in the textbook or online resource, you need to go back and re-examine your work. Make sure to identify the values of $u$ and $a$ correctly, substitute them into the formula, and simplify the expression to get the final answer. If you have made a mistake, you will need to correct it and try again.

Conclusion

In conclusion, the formula $\int u e^{a u} \, d u = \frac{1}{a^2}(a u - 1) e^{a u} + C$ is a powerful tool for solving integrals of exponential functions. By identifying the values of $u$ and $a$, substituting them into the formula, and simplifying the expression, we can find the integral of $x e^{5 x}$. We hope that this Q&A article has been helpful in answering some of the frequently asked questions about using the formula to solve integrals of exponential functions.