Eherre1989 3 years ago can someone solve this for me without changing the limits of integration?

1. Eherre1989

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

2. Eherre1989

3. Eherre1989

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

4. Mr.Math

Write what you have so far.

5. Eherre1989

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

6. Eherre1989

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

7. Eherre1989

i feel as if i should do by parts

8. Mr.Math

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

9. Mr.Math

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 $$-\cos^2\theta\sin\theta$$.

10. Eherre1989

i dont remember any trig identities

11. Mr.Math

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

12. Mr.Math

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

13. Eherre1989

oh ok thanks

14. Mr.Math

You're welcome.