Let’s use integration by parts:

$$ ∫ \cos(x)\cos(x)\,dx = \cos(x)\sin(x)-∫(-\sin(x))(\sin(x))\,dx $$If we apply integration by parts to the rightmost expression again, we will get $∫\cos^2(x)dx = ∫\cos^2(x)dx$, which is not very useful. The trick is to rewrite the $\sin^2(x)$ in the second step as $1-\cos^2(x)$. Then we get

\begin{align*} ∫ \cos^2(x)\,dx &= \cos(x)\sin(x)+∫(1-\cos^2(x))\,dx \\ &= \cos(x)\sin(x)+x-∫\cos^2(x)\,dx \end{align*}Now, all we have to do is to get $∫\cos^2(x)\,dx$ from the right-hand side to the left-hand side of the equation:

$$ 2∫ \cos^2(x)\,dx = \cos(x)\sin(x)+x $$ $$ ∫ \cos^2(x)\,dx = ÷{1}{2}(\cos(x)\sin(x)+x)+c $$One last question remains: where the hell did the $+c$ come from? I will answer that in a separate article in the future.