Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity

[yellowhappy]

1. Using Fourier Transforms to solve Definite Integrals with Limits 0 to Infinity

I'm trying to understand how to use Fourier Transforms to solve Definite Integrals with limits from 0 to Infinity.
I understand how to use Fourier Transforms to solve indefinite integrals, but I believe there is supposed to be a much simpler way to solve these definite integrals without working out the entire indefinite solution.

Homework Equations

X(j0) = Integral from -Infinity to Infinity : x(t)dt
x(0) = Integral from -Infinity to Infinity: X(jw)dw

 

[marcusl]

Welcome to PF! Here's a hint: Think about even and odd time functions.

 

[yellowhappy]

$$\displaystyle\int^\infty_0 2te^-^a^t\cos(t)\,dt = ?}$$

I'm not entirely sure how to manipulate this.. the $$e^-^a^t$$ makes it so the function is neither even nor odd anymore.. even if I pulled the even part out to rewrite the function so its integrated from -infinity to infinity I'm unsure about what to do with the odd piece.

 

[marcusl]

Yikes! I don't know what to do with this either. Sorry I jumped in...

 

[yellowhappy]

Yikes! I don't know what to do with this either. Sorry I jumped in...

I appreciate the thought! Thanks! :)