<s>[INST] Evaluate the integral $\int (\sin x + 2\cos x)^5 dx$ Do I have to expand manually $(\sin x + 2\cos x)^5$?
I don't think I can do integration by parts or substitution with this function.
Edit: Try to make one term, what I know is using $\sin^2 x + \cos^2 x=1$, but I think I still have to expand the polynomial.
 [/INST] One way is to recall the angle addition identity $$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.$$  If we suppose there exists a suitable constant $k$ and angle $\theta$ such that $$\sin x + 2 \cos x = k (\sin x \cos \theta + \cos x \sin \theta),$$ then we have $$k \cos \theta = 1, \\ k \sin \theta = 2,$$ hence $$k^2 (\cos^2 \theta + \sin^2 \theta) = 1^2 + 2^2 = 5,$$ hence $k = \sqrt{5}$ and we may take $\theta = \tan^{-1} 2$.  Then we have $$\int (\sin x + 2 \cos x)^5 \, dx = \int \left( \sqrt{5} \sin (x + \theta) \right)^5 \, dx = 5^{5/2} \int \sin^5 (x + \theta) \, dx.$$  The rest is handled by the usual methods.
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