Question

Convolution
\[sin\omega t * cos \omega t\]

Answer 1

\[sin\omega t * cos \omega t\]\[=\int_0^tsin\omega u \ cos \omega (t-u)du\]\[=-\frac{1}{\omega}\int_0^t\ cos \omega (t-u)d(cos\omega u)\]\[=\frac{1}{2\omega}[ cos ^2\omega (t-u)]_0^t\]
Doesn't seem right

Answer 2

\[ =\int_0^t \sin\omega u \; \cos \omega (t-u)du \\
=\int_0^t \sin\omega u \; (\cos \omega t \cos \omega u - \sin \omega t \sin \omega u)du \\
= \int_0^t \sin\omega u \; \cos \omega t \;\cos \omega u \;du - \int_0^t \sin \omega u \;\sin \omega t \; \sin \omega u \;du
\]

Answer 3

\[ =\int_0^t \sin\omega u \; \cos \omega (t-u)du \\
=\int_0^t \sin\omega u \; (\cos \omega t \cos \omega u - \sin \omega t \sin \omega u)du \\
= \cos \omega t \; \int_0^t \sin\omega u \;\cos \omega u \;du - \sin \omega t \;\int_0^t \sin \omega u \; \sin \omega u \;du \]