# How to integrate something that keeps on repeating

1. Nov 27, 2009

### dwilmer

1. The problem statement, all variables and given/known data
What is integral of e^t (1+3sint)?
I tried integ by parts but its not working
Im familiar with the technique where you end up with two idental integrals on either side, then you add one side to the other, and divide by 2.
This isnt working though. Mainly because the RHS will never be same as term on LHS, because the 1 either cancels or else becomes a t, at the first integration or differentiation.

2. Relevant equations

3. The attempt at a solution

2. Nov 27, 2009

### Staff: Mentor

You mean

$$e^t (1+3\sin(t))$$

or

$$e^{t (1+3\sin(t))}$$

not that I will know in either case, just it is not clear to me what the question is.

3. Nov 27, 2009

### HallsofIvy

Staff Emeritus
I suspect you mean $\int e^t (1+ 3sin(t)dt$ because Borek's suggested $\int e^{t(1+ 3sin(t)}dt$, while plausible from what you wrote, is intractible.

First, split the integral up: $\int e^t dt+ 3\int e^t sin(t)dt$. The first is easy and the second can be done by the technique you mention. One integration by parts gives a term of $\int e^t cos(t)dt$ and a second gives [itex]\int e^t sin(t)[/tex] again.

4. Nov 27, 2009

thanks Halls