Need help with Laplace transformations

[lucox]

I need help with calculation of several Laplace transformations. I'm not sure about following word expressions, which I'll use, as english is not my mother tongue, but I hope It will be understandable. 1. Find transformation to this object:
$$f(t) = 3t\sinh^2t - 4\int_{0}^{t}(e^s \cos hs - s^5e^3s - 1)ds$$
I got this result, but I'm not sure about it
$$F(p) = \frac{1}{p} - \frac{6p}{(p^2-1)^4} - 4(\frac{p-1}{(p-1)^2-1} - \frac{5}{(p-3)^6} - \frac{1}{p})$$2. Find transformation to this object:
$$f(t) = \int_{0}^{t}(4-6s\cos^23s+8\sin2s\cos2s)ds-3\frac{d}{dt}[te^{-2t}\sin3t+\cos^22t]$$3. Find source object f(t) related to this transformation:
$$F(p) = \frac{p+1}{p^2-2p+2}$$4. Find source object f(t) related to this transformation:
$$F(p) = \frac{5p(1-e^{-\frac{\pi}{2}p})}{(p+1)(p^2+4)}$$5. Find transformation of periodic function f(t), where
$$f_T(t)=\left\{\begin{array}{cc}3,&\mbox{ if } t\epsilon \left\langle0,\frac{\pi}{8}\right)\\\sin4t, & \mbox{ if } t\epsilon \left\langle \frac{\pi}{8},\frac{\pi}{2}\right)\end{array} T=\frac{\pi}{2}$$6. Find source object f(t) related to this transformation:
$$F(p) = e^{-\pi p} \frac{8}{p(p^2+4)} - \frac{2p^2-2p+8}{p(p^2+4)}$$
*** do I consider it right, that 1,2,5 are normal Laplace transformations and 3,4,6 are the inversed ones ?

Are they not too wild for homework or is this just my feeling ? I'm quite lost in them. Can you help me calculating and explaining the procedure ? I would appreciate any little effort on this. Is here somebody who could help me with this in next 3 hours ? I need to have it done until tomorrow. hmmm

BIG Thanks in advance

Luana

 

[gtoguy97]

1. Your answer is correct.2. The answer is:F(p) = \frac{4}{p} - \frac{6(p^2-1)}{(p^2-2p+2)^2} + \frac{8(p-1)}{(p^2-2p+2)^2} - \frac{3}{2(p^2+4)} + \frac{6}{(p^2+4)^2}3. The source object f(t) related to the transformation F(p) = \frac{p+1}{p^2-2p+2} is:f(t) = e^{-t}\sin\sqrt{2}t + \sqrt{2}e^{-t}\cos\sqrt{2}t4. The source object f(t) related to the transformation F(p) = \frac{5p(1-e^{-\frac{\pi}{2}p})}{(p+1)(p^2+4)} is:f(t) = 5\cos(\frac{\pi}{2}t) - 5e^{-\frac{\pi}{2}t}\cos(\frac{\pi}{2}t)5. The transformation of the periodic function f(t) is:F(p) = \frac{3}{p} + \frac{8}{(p^2+4)^2}6. The source object f(t) related to the transformation F(p) = e^{-\pi p} \frac{8}{p(p^2+4)} - \frac{2p^2-2p+8}{p(p^2+4)} is:f(t) = 8e^{-\pi t}\cos(\frac{\pi}{2}t) + (2t-1)e^{-\pi t}\sin(\frac{\pi}{2}t)