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I'm trying to do the following:

$$\int \cos(wt)w\sin(wt+\phi) \,\mathrm{d}t$$

I can't figure what to do here. I can't do integration by parts because of the $\phi$. Any ideas?

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    begin by using the addition identity on the sine function. Then this integral will yield to the usual techniques.2012-02-06
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    Note that $\cos(a) \sin(b) = (\sin(a+b)-\sin(a-b))/2$.2012-02-06
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    got it, that works2012-02-06
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    You could express it all with complex exponential functions. The integrand is $\frac{w}{4i}(e^{iwt}+e^{-iwt})(e^{\phi}e^{iwt}-e^{-\phi}e^{-iwt})$. At the end this might leave you with the issue of how to translate back to real functions, but that is not insurmountable.2012-02-06

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