Integrating Mix of Functions (Just Asking for Advice!)

Question

I'm asking for advice on how to integrate the following function:

$\int[x(\cos(x) + e^{2x})dx]$

My current progress: I know how to integrate both $\cos(x)$ and $e^{2x}$:

$\int[\cos(x)dx] = \sin(x)+C$

$\int[e^{2x}dx] = {e^{2x} \over 2}+C$

But I don't know where to go from there. Am I able to separate it into two integrals? Any help is appreciated!

Answer 1

Since you have a polynomial multiplied by a quantity that's easy to take the antiderivative of, use integration by parts, or the shorthand of tabular integration:

$$u = x\quad \mathrm dv = (\cos x+ e^{2x})\,\mathrm dx\quad \mathrm du = \mathrm dx\quad v = \sin x + \frac {e^{2x}}{2}$$