# Equivalent to \begin{align}\int \cos{\left(2x\right)} \ dx\end{align}?

For some reason, I am lost on one part of this integration problem:

\begin{align}\int \cos{\left(2x\right)} \ dx\end{align}

$u = 2x$

$du = 2 \ dx$

$\frac{1}{2} du = dx$

\begin{align}\frac{1}{2} \int \cos{\left(u\right)} \ du\end{align}

\begin{align}\frac{\sin{u}}{2} + C\end{align}

\begin{align}\frac{\sin{\left(2x\right)}}{2} + C\end{align}

The only issue is that I have been told that this should really equal $\sin{x}\cos{x} + C$. I know that $\sin{\left(2x\right)} = 2\sin{x}\cos{x} + C$, so:

\begin{align}\frac{\sin{\left(2x\right)}}{2} = \frac{2\sin{x}\cos{x}}{2} + C = \sin{x}\cos{x} + C\end{align}

On a question I had asked previously, I was told that \begin{align} \int \cos{\left(2x\right)} \ dx \neq \frac{\sin{\left(2x\right)}}{2} + C\end{align}, but rather $\sin{x}\cos{x} + C$.

Please excuse the silly question, but these two expressions are the same, right?

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They are equal. Why didn't you link to the old question? –  anon Mar 14 '12 at 6:39
The two answers are the same. –  André Nicolas Mar 14 '12 at 6:41
@anon You'd have to dig around to find what I'm references. –  Oliver Spryn Mar 14 '12 at 6:42
Thank you all!! –  Oliver Spryn Mar 14 '12 at 6:42

$$\int \cos(2x)dx=\frac{\sin(2x)}{2}+C=\sin(x)\cos(x)+C$$
That is because $sin(2x) = 2 sin(x) cos(x)$, which is what he needs to know explicitly –  Jeremy Carlos Mar 14 '12 at 11:46