hi I'm having trouble integrating this
$\displaystyle \int sin^3\theta\cdot 3sin\theta cos\theta$
I don't really have a method for integrating trig it's normally pretty simple to work out but i don't know where to start with this one :/
ok so this question actually has limits $\displaystyle \int_{0}^{\frac{\pi}{2}}$
now i have the mark scheme for this question which shows that the limits stay the same. But i thought when you did integration by substitution you had to manipulate the limits :/
You have two possibilities, while making a substitution :
- you substitute, and you change the boundaries
- you substitute, and you don't change the boundaries. In which case, you have to «back substitute», that is : after you got an antiderivative in terms of the new variable, substitute it with the old variable, and keep the limits.
But I think the first one is better, because for the second one, it would be more correct to write :
$\displaystyle \int_{\theta=0}^{\theta=\pi/2} u^4 ~du$, so that you know that it's the boundaries for theta.
While for the first situation, you can just say $\displaystyle \int_0^1 u^4 ~du$
Got it ?
$\displaystyle \theta=0 \Rightarrow u=\sin\theta=0$
$\displaystyle \theta=\pi/2 \Rightarrow u=\sin\theta=1$
Be careful of the order of the boundaries.
In the case where you let $\displaystyle y=\cos\theta$, you would have $\displaystyle \int_1^0 =-\int_0^1$
But that's another story ^^
Thanks!
would i integrate this by substitution?
$\displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}} sin^22\theta d\theta$
or change into compound angle i.e.
$\displaystyle sin2\theta$ = $\displaystyle 2sin\theta cos\theta$ and then square it?
or does the square prevent this?