Calculate $\int e^{2x}(\cos x)^3 dx$ 
Calculate $$\int e^{2x}(\cos x)^3 dx$$

My try:


*

*Firsty I tried to use integration by parts but then I got:
$$\int e^{2x}\cos^3(x) dx=...=\frac{1}{2}\cos^3(x) e^{2x}+\frac{3}{2}\left(\int e^{2x} \sin(x) dx-\int e^{2x} \sin^3(x) dx \right)$$So my calculation have not many sens because I have $\int e^{2x} (\sin x)^3 dx $ so I returned to similar problem how in the task.

*After that I tried to use integration by substitution:
$$u=\cos(x)$$
$$du=-\sin(x)dx$$
However in this way I don't know how to transform $e^{2x}$ depending on the $u$.
Have you any idea how to do this task?


Linearise $\cos^3 x$ first: $\;\cos 3x=4\cos^3x-3\cos x$, so $\;\cos^3x=\frac14(\cos 3x+3\cos x)$, whence
$$\mathrm e^{2x}\cos ^3 x=\tfrac14\operatorname{Re}\Bigl(\mathrm e^{(2+3i)x}+3\mathrm e^{(2+i)x}\Bigl)$$
so calculate $\;\frac14\displaystyle\int\bigl(\mathrm e^{(2+3i)x}+3\mathrm e^{(2+i)x}\bigl) \mathrm dx$ and take its real part.
