TITLE: Calculate $\int e^{2x}(\cos x)^3 dx$
QUESTION [4 upvotes]: 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?

REPLY [2 votes]: $$\cos^3 x = \dfrac{1}{4}\left(\cos(3x)+3\cos x\right)$$
Plugging in gives:
$$\int e^{2x}\cos^3 x dx = \dfrac{1}{4}\int e^{2x}\cos(3x)dx + \dfrac{3}{4}\int e^{2x}\cos x dx$$
Looking at the first integral and performing IBP twice:
$$\int e^{2x}\cos(3x)dx = \dfrac{2}{13}e^{2x}\cos(3x)+\dfrac{3}{13}e^{2x}\sin(3x)$$
Then, the second integral and performing IBP twice:
$$\int e^{2x}\cos x dx = \dfrac{2}{5}e^{2x}\cos x + \dfrac{1}{5}e^{2x}\sin x$$
So, returning to the initial problem:
$$\int e^{2x}\cos^3 x dx = e^{2x}\left(\dfrac{1}{26}\cos(3x)+\dfrac{3}{52}\sin(3x) + \dfrac{3}{10}\cos x + \dfrac{3}{20}\sin x\right)+C$$
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