# Difficult integral by sub., parts & table

• silicon_hobo
In summary, the conversation was about the integral \int xe^{2x^2} cos(3x^2) dx and the attempt at solving it using the formula \int e^{au} cos\ bu\ du\ = \frac{e^{au}}{a^2+b^2}(a\ cos\ bu\ + b\ sin\ bu)+C. The user shared their solution and asked for help in submitting scanned work on the forum.
silicon_hobo

## Homework Statement

$$\int xe^{2x^2} cos(3x^2) dx$$
This is the hardest integral I've attempted so far. I've come up with an answer that fits a table in my book but I'm not sure if I arrived there correctly. Thanks for reading!

## Homework Equations

$$\int e^{au} cos\ bu\ du\ = \frac{e^{au}}{a^2+b^2}(a\ cos\ bu\ + b\ sin\ bu)+C$$

## The Attempt at a Solution

http://www.mcp-server.com/~lush/shillmud/int2.7.JPG
http://www.mcp-server.com/~lush/shillmud/int2.72.JPG

substitute:

$$u=x^2$$

$$du=2x dx$$

$$I=\int xe^{2x^2}\cos(3x^2){\rm dx}\,=\, {1\over 2} \int e^{2u}\cos(3u) {\rm du}$$

I have done some shortcut here, with use of the formula, your number two:

$$I={1\over 26} e^{2u} (2 \cos(3u)\,+\,3\sin(3u))\,+\,C$$

$$I={1\over 26} e^{2x^2} (2 \cos(3x^2)\,+\,3\sin(3x^2))\,+\,C$$

this is the same as integrator...

http://integrals.wolfram.com/

check it out

Here's the solution, I didn't simplify it any more like I should have, but I hope you understand the basic process. I checked my answer with CAS and it's correct, so don't worry about that.
http://img411.imageshack.us/img411/5615/file0001su6.jpg

Last edited by a moderator:
How do you submit scanned work onto here?
Thanks

SavvyAA3 said:
How do you submit scanned work onto here?
Thanks

Assuming the scanned image is in a format the forum recognises, jpeg,mpeg and so on either click on the icon in yellow with a black mountain or two and a black sun, visible in the new reply or go advanced: edit windows, not quick post. Or type [noparse]*url [/noparse] around the files url.

To get a url for an image stored on your hard drive you will need to upload it to a web host facility such as photobucket or freewebs. And then cut and paste the url. If not then use the paper clip to attach files directly from your hard drive to the forum. With this option there will be a delay while the forum scans the file for viruses, but any format that is listed there from jpeg to gif are acceptable.

Last edited by a moderator:
Thanks alot!

## 1. What is the purpose of using integration by substitution?

Integration by substitution is a technique used to solve integrals that cannot be solved by traditional methods. It involves replacing a complicated expression with a simpler one in order to make the integral easier to evaluate.

## 2. How do I choose the substitution for a difficult integral?

The substitution chosen for a difficult integral should be based on the form of the integral. Common substitutions include trigonometric functions, exponential functions, and u-substitutions. It is important to choose a substitution that will simplify the integral and make it easier to solve.

## 3. What is the difference between integration by parts and integration by substitution?

Integration by parts and integration by substitution are both techniques used to solve integrals, but they differ in the approach. Integration by parts involves using the product rule of differentiation, while integration by substitution involves using the chain rule of differentiation.

## 4. Is there a specific order in which I should use integration by parts and integration by substitution?

There is no specific order in which these techniques should be used. It is important to consider the form of the integral and choose the method that seems most appropriate. In some cases, a combination of both methods may be necessary to solve a difficult integral.

## 5. What is the role of the table of integrals in solving difficult integrals?

The table of integrals provides a list of known integrals and their corresponding solutions. It can be used as a reference when solving difficult integrals, as it may contain the solution to the integral in question. However, it should not be solely relied upon and proper understanding of integration techniques is still necessary.

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