$\mathcal {L}(\operatorname e^{-6t}\cos(5t))=?$

Question

Find the laplace transform of the following equation $f(t) =\operatorname e^{-6t}\cos (5t)$

Answer 1

Hint: By one of the shift theorems, you have $\mathcal{L}\{e^{at}f(t)\} =F(s-a)$. In your problem, we would then take $f(t)= \cos(5t)$. Thus, $$\mathcal{L}\{e^{-6t}\cos(5t)\} = \left.\mathcal{L}\{\cos(5t)\}\right|_{s\to s+6}=\ldots$$

In the end, you should get $\mathcal{L}\{e^{-6t}\cos(5t)\} = \dfrac{s+6}{(s+6)^2+25}$.

Answer 2

try writing $cos 5t$ as $\frac12 (e^{5it} + e^{-5it})$

Answer 3

ok let us use following formula for laplace transform

http://upload.wikimedia.org/math/0/c/9/0c9c50441866da37706842c765ad9859.png

after that we get following one

$$\int(cos(5t)*e^{-t(6+s)})dt$$

according to this

http://www.wolframalpha.com/input/?i=int%28cos%285t%29*e%5E%28-t%286%2Bs%29%29+from+-infinity+to+plus+infinity

this does not converge,you can use also formulas from this table

http://eeweb.poly.edu/~yao/EE3054/FormulaSheets_Test2.pdf

EDITED: there is picture