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,
 A: 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.
A: You can also use complex numbers to make it mechanical.
Write $\displaystyle \cos x = \dfrac{e^{ix} + e^{-ix}}{2}$ and just expand out.
This method makes finding integrals of the form
$$\int P(\sin x, \cos x) \ \text{dx}$$ 
(where $P(x,y)$ is a polynomial in the variables $x,y$) purely mechanical, without having to do any clever algebra etc.
A: Expand $(1+\cos(x))^3$. And rewrite $\cos^n(x)$ in terms of $\cos(nx)$ and other similar terms.
$(1+\cos(x))^3 = 1 + 3 \cos(x) + 3 \cos^2(x) + \cos^3(x) = \frac{\cos(3x)+6 \cos(2x)+15 \cos(x)+10}{4}$.
Now, integrate term by term to get the desired answer.
$\int \cos(nx) = \frac{\sin(nx)}{n}$.
Hence, $\int (1+\cos(x))^3 = \frac{\frac{\sin(3x)}{3}+6\frac{\sin(2x)}{2}+15 \sin(x) + 10x}{4} + c = \frac{\sin(3x)}{12} + \frac{3\sin(2x)}{4} + \frac{15 \sin(x)}{4} + \frac{5x}{2} + c$.
A: HINT:
You can use the Half-angle formula. $$\cos{2x} = 2\cos^{2}{x}-1$$ and so you have $$1+\cos{x}= 1 + 2\cos^{2}\frac{x}{2} -1 =2\cos^{2}\frac{x}{2}$$ So the required integral is 
\begin{align*}
\int (1+\cos{x})^{3} \rm{dx} &= \int 8 \cdot \cos^{6}\Bigl(\frac{x}{2}\Bigr) \ \rm{dx} \\ &= 8 \int \cos^{6}\Bigl(\frac{x}{2}\Bigr) \ \rm{dx}
\end{align*}
