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.
Best Answer
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.