# Help with integral

by jaygatsby
Tags: integral
 P: 3 This is not a homework problem, but a problem in the textbook that is not required. I am doing this to get a handle on the topic. I am evaluating a Fourier transform, without tables, and need to evaluate this integral: $$\int e^{-t} * sin(2 \pi f_c t) * e^{-j2 \pi ft} dt$$ I have tried two methods: 1) integration by parts, and 2) integration after expressing the sine function as a complex exponentials. I get stuck in both cases. The asterisks are there to assist with clarity of spacing. Thanks for any help you can provide, J
 Sci Advisor P: 5,768 Use Euler formula to get exp(-t)*trig function. This is a standard integral (find in table). Trig function: sin(at), integral = a/(1 + a2) cos(at), integral = 1/(1 + a2) (a > 0 for both)
 P: 3 Thanks, I did try Euler's formula but then worked the integral out manually (attempted to...) So this integral I would find in the table exclusively, and never try without a table? The way the drill is stated in the book (not a homework problem.), I wonder if I am to work it out without a table. Thanks, J
P: 5,768

## Help with integral

You can integrate by parts twice to get an equation involving the original integral.

I(exp(-t)cos(at)) = 1 + aI(exp(-t)sin(at)) = 1 - a2I(exp(-t)cos(at))

Similarly for sin(at) integral.
 P: 3 Thank you

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