Integrate this function of $\theta$ with respect to $x$? I have a definite integral which is confusing me:
$$ \int_{0}^{\frac{\pi}{2}} sin(3\theta)cos(\theta) dx $$
I tried to fiddle with a substitution of $ x=sin(\theta) $ and tried to go about the question as I would do with a u-substitution question but I got nowhere. Then that got me thinking: what am I supposed to do to integrate something like this where the variable of the function I'm integrating is different to what I'm integrating with respect to? 
Also, another side-question is: if the method for this is not substitution, why? Why isn't there some kind of substitution for this?
I should probably say that whilst trying to find an answer on how to do this, I stumbled upon someone saying that you should take the stuff with theta out in front of the integral and treat it as a constant and then integrate 1 w.r.t. x but I don't know if this is useful in this case, since this is a definite integral.
Also, the answer on my sheet is $\frac{1}{2} $, if that's of any use.
 A: If this problem is stated correctly then there are no fancy tricks needed; you are overthinking this one. $ \sin(3\theta)\cos(\theta)$ is a constant with respect to $x$. So yes it can be pulled out of the integral. As $x$ changes, $\sin(3\theta)\cos(\theta)$ will not change. An analog to this problem would be to find the derivative of $e^\pi$. A knee-jerk reaction might make some say the derivative must be $e^\pi$ since $e^x$ is its own derivative. But $e^\pi$ is a constant, and so the derivative is just $0$. Can you integrate $$ C\int_{0}^{\frac{\pi}{2}} dx $$ where $C$ is any constant? Try on a few trig identities for your constant and see if you can get some simplifications.
A: You need not use any substitution for the function as it is independent of $x$
Notice, given that $$\int_{0}^{\pi/2} \sin(3\theta)\cos(\theta)\ dx$$ 
we see that the function $\sin(3\theta)\cos(\theta)$ is being integrated w.r.t. $x$ from $x=0$ to $x=\pi/2$ hence $x$ is variable which does not depend on the function of $\theta$ hence $\sin(3\theta)\cos(\theta)$ is treated as a constant so we have $$\int_{0}^{\pi/2} \sin(3\theta)\cos(\theta)\ dx=\sin(3\theta)\cos(\theta)\int_{0}^{\pi/2} \ dx$$$$=\sin(3\theta)\cos(\theta)[x]_{0}^{\pi/2}$$$$=\sin(3\theta)\cos(\theta)\left[\frac{\pi}{2}-0\right]$$ $$=\color{red}{\frac{\pi}{2}\sin(3\theta)\cos(\theta)}$$
A: The trick is to use the product-to-sum formula:
$$
\sin(3\theta)\cos(\theta)=\frac{1}{2}\left(\sin(4\theta)+\sin\theta\right).
$$
Now the integral splits up into two pieces; the first vanishes, and the second gives you $1/2$.
A: Hint:
Let $\simeq$ denote the equality "up to" some constant; then here is yet another way:
$$
\int_{\theta} \sin 3\theta \cos \theta = \int_{\theta} \sin 3\theta D\sin \theta \simeq \sin \theta \sin 3\theta - 3\int_{\theta} \sin \theta \cos 3\theta;\\
-3\int_{\theta} \sin \theta \cos 3\theta = 3\int_{\theta} \cos 3\theta D\cos \theta \simeq 3\cos \theta \cos 3\theta + 9\int_{\theta} \cos \theta \sin 3\theta,
$$
so 
$$
-8 \int_{\theta} \sin 3\theta \cos \theta \simeq 3\cos \theta \cos 3\theta + \sin \theta \sin 3\theta;
$$
this gives $1/2$ as claimed. But the integral $\int_{x=0}^{\pi/2} \sin 3\theta \cos \theta = \frac{\pi}{2}\sin 3\theta \cos \theta$.  
