# Solving the Integral of 3e^(t/3)*cos(3t): Help Needed!

• ns5032
In summary: So just be careful with that.In summary, the integral of [3e^(t/3)]*cos(3t) can be solved by using integration by parts twice. The first time, u = e^(t/3) and dv = cos(3t), and the second time, u = e^(t/3) and dv = sin(3t). This will result in the original integral being equal to a negative constant times itself, plus some other terms. By rearranging the equation and dividing by the constant, the final answer for the original integral is equal to the remaining terms divided by 2.
ns5032
The integral I need help with is:

integral of:

[3e^(t/3)]*cos(3t)

I know that I can pull out the 3 in front of the integral, but after that what??

Do you know how to http://en.wikipedia.org/wiki/Integration_by_parts"

Last edited by a moderator:
$$3\int e^{\frac t 3}\cos{3t}dt$$

You need to show more effort than that if you want to receive a lot of help.

If I'm thinking of the problem correctly, you can solve it by doing integration by parts twice -- the second time you do it you should receive your initial integral again and some other terms (if you made the right substitutions for your u and v terms). Then you just solve for your integral and you're done. I didn't work it out at all, so that could be wrong. However, I think that's usually the case when you have an exponential and a single trig term.

Well if I do integration by parts, the cos switches back and forth between sin and cos, and the e^(t/3) stays in there every time... I don't know what I'm doing wrong...

u = e^(t/3)
du = (1/3)e^(t/3)

v = (1/3)sin(3t)
dv = cos(3t)

so then it's:
uv- integral of v*du
[e^(t/3)*(1/3)sin(3t)]-integral of [(1/3)sin(3t)*(1/3)e^(t/3)]
and by simplifying:
[(1/3)e^(t/3)*sin(3t)]-(1/9)*integral of [sin(3t)*e^(t/3)]

It just seems like I'll keep making circles...?

ns5032 said:
Well if I do integration by parts, the cos switches back and forth between sin and cos, and the e^(t/3) stays in there every time... I don't know what I'm doing wrong...

It just seems like I'll keep making circles...?

Have faith -- it's supposed to do that! Integrals of the form exponential times sine or cosine have to go through two levels of integration by parts. What you'll see, after the second time, is that you get, on the right-hand side of the equation, your original integral back multiplied by a negative constant. You now have an exercise in algebra! Move that multiplied integral to the other side and you'll have an expression which reads (constant) times original integral = remaining right-hand side terms. If you divide through by the constant, you'll have the result for your original integral (+C).

you will need to make one circle to solve the integral. like i said this will give you your original integral and some other terms. it will look like this:

OrigIntegral = -OrigIntegral + Terms
2OrigIntegral = Terms
OrigIntegral = Terms/2, the final answer.

You need to make sure that the 'Terms' part doesn't get thrown out in your second integration by parts attempt. This will happen if you chose the wrong u and v the second time around, in which case you will be left with:

## 1. How do I approach solving this integral?

To solve this integral, you can use the method of integration by parts. This involves identifying the parts of the integral that can be differentiated and integrated separately.

## 2. What should I do with the 3e^(t/3) term?

The 3e^(t/3) term can be treated as a constant and can be factored out of the integral. This will simplify the integration process.

## 3. How do I integrate the cos(3t) term?

To integrate the cos(3t) term, you can use the trigonometric identity cos(3t) = (e^(3it) + e^(-3it))/2. This will allow you to rewrite the integral in terms of e^t, which can be easily integrated.

## 4. What is the final answer to this integral?

The final answer to this integral will be in the form of a constant multiplied by e^(t/3)*sin(3t). This can be confirmed by taking the derivative of the answer and checking if it matches the original function.

## 5. Are there any special cases or exceptions to consider when solving this integral?

Yes, when solving this integral, you must be careful to include the constant of integration. Additionally, if the limits of integration are given, you must substitute these values into the final answer. Also, check for any potential restrictions on the domain of the original function.

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