# Thread: Semi urgent - Laplace Transform

1. ## Semi urgent - Laplace Transform

Hi all,

I'll state the question, then explain my thinking.

Evaluate $\displaystyle \int_0^{\inf} [ H(t - \frac{\pi}{4}) - H(t)] \cos{2t}e^{-st} dt$ where $\displaystyle s > 0$

Ok, I know the heaviside function limits the range between 0 and $\displaystyle \frac{\pi}{4}$.

I also know that the laplace transform of $\displaystyle \cos{2t}$ is $\displaystyle \frac{s}{s^2 + 4}$

What I don't know is how I'm meant to effectively combine these properties together to correctly evaluate the integral.

Any help would be greatly appreciated.

2. Your integral is equal to $\displaystyle \int_0^{\pi/4}- \cos(2t)\,e^{-st} \,\mathrm{d}t$ is it not?

3. I can see how the heaviside function can produce the new limits, however I can't see how you got the negative out the front of the variable and I'm unsure of what to do with it from there.

4. Actually, I think I use this from the table of integrals to produce the answer from the above...

But again I'm unsure how you got that negative out the front.

5. What value does $\displaystyle H(t-\pi/4) - H(t)$ have on the interval [0,pi/4]?

From there, use integration by parts I think.

6. Ah ok, -1 obviously. Thanks.