How can I find $ \int{(1 + \cos x)^3\mathrm dx} $? My question is ; How can I solve the following integral question?
$$\int{(1 + \cos x)^3\mathrm dx}$$
Thanks in advance,

If you don't know how to rewrite $\cos^n(x)$ in terms of $\cos(nx)$, here is an alternate solution.
Expand $(1+\cos(x))^3 = 1 + 3\cos(x) + 3\cos^2(x) + \cos^3(x)$.
Now, you know how to integrate the first two terms.
The third term can be integrated by using the half-angle formula
$$\cos^2(x) = \frac{1+\cos(2x)}{2},$$
so that
$$\int 3\cos^2(x)\,dx = 3\left(\frac{x}{2} + \frac{\sin(2x)}{4}\right)$$
The last term can be integrated using the pythagorean identity $\cos^2(x) = 1 - \sin^2(x),$ so
$$\cos^3(x) = \cos(x) - \cos(x)\sin^2(x).$$
And now $\int \cos(x)\,dx = \sin x$ and $\int \cos(x)\sin^2(x)\,dx = \frac{\sin^3(x)}{3}$, so
$$\int \cos^3(x)\,dx = \sin x - \frac{\sin^3(x)}{3}.$$
Now just put the pieces together.
