# Laplace Transform/Examples/Example 1

## Example of Laplace Transform

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

$\laptrans {4 e^{5 t} + 6 t^3 - 3 \sin 4 t + 2 \cos 2 t} = \dfrac 4 {s - 5} + \dfrac {36} {s^4} - \dfrac {12} {s^2 + 16} + \dfrac {2 s} {s^2 + 4}$

## Proof

 $\ds \laptrans {4 e^{5 t} + 6 t^3 - 3 \sin 4 t + 2 \cos 2 t}$ $=$ $\ds 4 \laptrans {e^{5 t} } + 6 \laptrans {t^3} - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}$ Linear Combination of Laplace Transforms $\ds$ $=$ $\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \laptrans {t^3} - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}$ Laplace Transform of Exponential $\ds$ $=$ $\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}$ Laplace Transform of Positive Integer Power $\ds$ $=$ $\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \paren {\dfrac 4 {s^2 + 4^2} } + 2 \laptrans {\cos 2 t}$ Laplace Transform of Sine $\ds$ $=$ $\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \paren {\dfrac 4 {s^2 + 4^2} } + 2 \paren {\dfrac s {s^2 + 2^2} }$ Laplace Transform of Cosine $\ds$ $=$ $\ds \dfrac 4 {s - 5} + \dfrac {36} {s^4} - \dfrac {12} {s^2 + 16} + \dfrac {2 s} {s^2 + 4}$ simplifying

$\blacksquare$