Help Evaluating Integral with Euler's Formula

[jaygatsby]

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

 

[mathman]

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)

 

[jaygatsby]

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

 

[mathman]

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.

 

[jaygatsby]

Thank you