Problem 04 Evaluate $\displaystyle \int_0^\infty \dfrac{e^{-t}\sin t}{t} ~ dt$.
Solution 04 From Problem 01 | Division by t: $\mathcal{L} \left( \dfrac{\sin t}{t} \right) = \arctan \left( \dfrac{1}{s} \right)$
By first shifting property:
Problem 03 Find the value of $\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt$
Problem 04 Find the Laplace transform of $f(t) = e^t \sinh 2t$.
Solution 04
Problem 03 Find the Laplace transform of $f(t) = e^{-3t} \cos t$.
Solution 03
Problem 02 Find the Laplace transform of $f(t) = e^{-5t} \sin 3t$.
Solution 02
Problem 01 Find the Laplace transform of $f(t) = e^{2t}t^3$.
Solution 01
First Shifting Property If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, when $s > a$ then,
In words, the substitution $s - a$ for $s$ in the transform corresponds to the multiplication of the original function by $e^{at}$.
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