# Solving integrals with the table of integrals

• Jgoshorn1
In summary, the problem is to evaluate the integral of e2xarctan(ex)dx. According to the table of integrals, it can be solved using either formula #92 or #95. After attempting to make a substitution with e^x, the solution is found to be 1/2(e2x+1)arctan(ex) - (1/2)ex + C.
Jgoshorn1

∫e2xarctan(ex)dx

## Homework Equations

From the table of integrals:
#92 ∫utan-1udu = (u2+1)/2)tan-1-u/2 + c

or

#95 ∫untan-1udu = 1/(n+1)[un+1tan-1-∫ (un+1du)/(1+u2) , n≠-1

## The Attempt at a Solution

The answer is 1/2(e2x+1)arctan(ex) - (1/2)ex + C

I don't know if I'm supposed to make a substitution first and if so what I should substitute and/or if which from I should use from the table. I've tried to make the initial substitution of e2x and ex and they both got me seemingly nowhere. Help please.

Jgoshorn1 said:

∫e2xarctan(ex)dx

## Homework Equations

From the table of integrals:
#92 ∫utan-1udu = (u2+1)/2)tan-1-u/2 + c

or

#95 ∫untan-1udu = 1/(n+1)[un+1tan-1-∫ (un+1du)/(1+u2) , n≠-1

## The Attempt at a Solution

The answer is 1/2(e2x+1)arctan(ex) - (1/2)ex + C

I don't know if I'm supposed to make a substitution first and if so what I should substitute and/or if which from I should use from the table. I've tried to make the initial substitution of e2x and ex and they both got me seemingly nowhere. Help please.

Try ##u=e^x## and see if you can't get it in a form to use 95 with ##u^n## in front for some ##n##.

## 1. How do I use the table of integrals to solve an integral?

The table of integrals is a reference tool that lists common integrals and their corresponding solutions. To use it, find the integral you want to solve in the table and identify the corresponding formula. Then, you can apply the formula to your specific integral to find the solution.

## 2. What are the benefits of using the table of integrals to solve integrals?

The table of integrals can save time and effort in solving integrals, as it provides a quick reference for common integrals and their solutions. It can also help to check the accuracy of your solution by comparing it to the listed solution in the table.

## 3. Can I use the table of integrals for any type of integral?

The table of integrals contains a wide range of common integrals, but it may not have solutions for every type of integral. It is best used for integrals that can be solved using common integration techniques such as substitution or integration by parts.

## 4. How do I know which formula to use from the table of integrals?

The formula you use from the table of integrals will depend on the form of your integral. For example, if your integral contains a trigonometric function, you would use the corresponding trigonometric formula from the table. If you are unsure, it may be helpful to consult a calculus textbook or ask a math tutor for guidance.

## 5. Can I memorize the table of integrals instead of using it as a reference?

While it is not necessary to memorize the entire table of integrals, it can be helpful to memorize some of the most common formulas. This can save time in solving integrals and also serve as a useful tool for checking the accuracy of your solutions.

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