can someone solve this for me without changing the limits of integration?

Question

anonymous · Answer

$$\int\limits_{1}^{3}x \sqrt{1-(x-2)^2}dx$$

anonymous · Answer

step by step please

anonymous · Answer

it has to be a trig substitution i just get stuck when i have to integrate the sins

mr.math · Answer

Write what you have so far.

anonymous · Answer

$$\int\limits_{}^{}(\sin \theta-\sin^3\theta-2\sin^2\theta+2)d \theta$$

anonymous · Answer

i need that integrated i cant seem to figure out the middle two

anonymous · Answer

i feel as if i should do by parts

mr.math · Answer

Break it down into four integrals. I'm sure you know how to integrate $\sin\theta$ and $2$.

mr.math · Answer

For $\sin^3\theta$, write $\sin^3\theta=\sin\theta-\cos^2\theta\sin\theta$, and then the integral of $\sin\theta$ is easy. Substitute $u=\cos\theta$ for the integration of $-\cos^2\theta\sin\theta$.

anonymous · Answer

i dont remember any trig identities

mr.math · Answer

$\sin^2\theta=\frac{1-\cos(2\theta)}{2}$.

mr.math · Answer

Sorry it should be $-2\sin^2\theta=\cos(2\theta)-1$.

anonymous · Answer

oh ok thanks

mr.math · Answer

You're welcome.