Find Laplace transform of $(sin^2 2t)$

Question

Find Laplace transform of $(\sin^2 2t)$

How do I go about this ?

do I spilt them up like

$ L ( \sin 2t) \cdot L (\sin 2t) $ ?

Answer 1

No! The Laplace transform of fg is NOT L(f)L(g). Instead use the basic definition: $\int_0^\infty sin^2(2t)e^{-st}dt$. You might find it simplest to use the fact that $sin(x)= \frac{e^{ix}- e^{-ix}}{2i}$ so that $sin(2t)= \frac{e^{2it}- e^{-2it}}{2i}$ and $sin^2(2t)= \left(\frac{e^{2it}- e^{-2it}}{2i}\right)^2= -\frac{e^{4it}- 2+ e^{-4it}}{4}$.

Answer 2

Hint: $$\sin^2 2t= \dfrac{1-\cos 4t}{2}$$

Answer 3

Given $\sin^22t$

$$\sin^22t=\dfrac{1-\cos4t}{2}$$ $$L(\sin^22t)=\dfrac{L(1)-L(\cos4t)}{2}=\dfrac{1}{2s}-\dfrac12\left(\dfrac{s}{s^2+16}\right)$$