I am trying to find $\int_{0}^{t}{(g(\tau)\sin(\omega_{0}t - \tau))d\tau}$ for an arbitrary $g(\tau)$, resulting from using a convolution method to solve the differential equation.

I usually deal with complicated integrals involving sin or cos functions by using an integration by parts, but here it does not work, because the second integral of $g(\tau)$ is not necessarily equal to $g(\tau)$. It seems integration by substitution is not possible here as well.

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    $\begingroup$ I think there is no closed form. $\endgroup$ – Alex Silva Nov 21 '16 at 19:41
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    $\begingroup$ One could write this in terms of the Fourier Transform of $g$. $\endgroup$ – Mark Viola Nov 21 '16 at 19:43
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    $\begingroup$ What do you mean "find"? There is no closed form for arbitrary $g(t)$ unless you accept some transforms and its inverts (like Fourier transform) $\endgroup$ – Serge P. Nov 22 '16 at 10:32

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