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I am struggling to evaluate this integral: $$ \int e^{(\mu +r)t+\frac{r}{\alpha}\sin(\alpha t)}\cos(\alpha t)dt $$ The integration just gets harder and harder when integrating by parts! Any tips?

Thanks in advance!

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  • $\begingroup$ Which equation? $\endgroup$ – José Carlos Santos Mar 25 '19 at 17:27
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    $\begingroup$ Make sure you're choosing the right parts. What's your $u$ and $dv$? $\endgroup$ – Michael Biro Mar 25 '19 at 17:31
  • $\begingroup$ Wolfram Alpha cannot solve analytically even the greatly simplified case of $\mu=0,r=\alpha=1$. Is this an expression you arrived at on your own? $\endgroup$ – jawheele Mar 25 '19 at 19:04
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Hint: write $$e^{(\mu+r)t+\frac r\alpha\sin(\alpha t)}\cos(\alpha t)=e^{(\mu+r)t}e^{\frac r\alpha\sin(\alpha t)}\cos(\alpha t)$$Notice that the last two terms have trigonometric functions with the argument $(\alpha t)$. This indicates there may be a connection between these two terms. What happens if you integrate $e^{\frac r\alpha\sin(\alpha t)}\cos(\alpha t)$?

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