Integration of sqrt(4-x^2)/x*dx

[heartshapes]

If we let x=2sinx then the integration of the sqrt(4-x^2)/x*dx on [1,2] is equivalent to ....

I have no idea. I tried so many different things and I didn't get an answer that was one of the choices..

(a) the integration of cos^2x/sinx on [0,2]
(b) the integration of cosx/sinx on [pi/6,pi/2]
(c) 2 * the integration of cos^2x/sinx on [pi/6,pi/2]
(d) the integration of cosx/sinx on [1,2]
(e) none of the above.

So I am thinking that its none of the above.. but my work makes no sense.

Any help would be greatly appreciated. Thank you!!

 

[stapel]

If we let x=2sinx then....

If x = 2sin(x), then x is some constant value, that value being the solution to this equation.

Eliz.

 

[galactus]

If you let \(\displaystyle x=2sin({\theta}), \;\ dx=2cos({\theta})d{\theta}\) and make the subs you get:

\(\displaystyle \frac{\sqrt{4-4sin^{2}({\theta})}}{2sin({\theta})}2cos({\theta})d{\theta}\)

\(\displaystyle =2\int\frac{cos^{2}({\theta})}{sin({\theta})}d{\theta}\)

Change the limits of integration.

\(\displaystyle 1=2sin({\theta}); \;\ {\theta}=sin^{-1}(\frac{1}{2})=\frac{\pi}{6}\)

\(\displaystyle 2=2sin({\theta}); \;\ {\theta}=sin^{-1}(1)=\frac{\pi}{2}\)

Does of one your choices match that?.

 

[heartshapes]

YES! Thank you so much!