# Integration using parts

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?

• Make sure you're choosing the right parts. What's your $u$ and $dv$? Mar 25 '19 at 17:31
• 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? Mar 25 '19 at 19:04
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 $$e^{\frac r\alpha\sin(\alpha t)}\cos(\alpha t)$$?