You're on the right track.

Now differentiate that. We want dO. I'm using capital letter o to denote theta because I have no idea how to put it into my computer.

dx = 2cos O dO

Now to get rid of the other x's. Substitute all of the x's with the 2sin O and simplify. The denominator may be the hardest part. Once you have that, you should end up with:

int (4sin^2 O 2cos O dO / 2cos O)

Do the algebra:

4 int (sin^2 O) dO

Use a power reducing identity, evaluate the integral, and make sure you substitute the x's back in! You can get the solution on your own. Can't do all of the work, can I?

Miss Kristin

theta=v.

x=2sinv

dx=2cosv

integral[((cos^2v)(2cosv))/(sqrt(4-4sin^2v)) *dv]

integral[(2cos^3v)/(2cosv) * dv]

integral[cos^2v]

cos^2v=1/2(1+cos2v)

1/2*integral[(1+cos2v)dv]

1/2[v+(sin2v)/2]

x=2sinv

x/2=sinv

sin^-1(x/2)=v

now sin2v=2sinvcosv

(2sinvcosv)/2=sinvcosv. to find cosv draw a triangle. cosv=sqrt(4-x^2)/2 and sin v=x/2

1/2[sin^-1(x/2) + (xsqrt(4-x^2))/4] + C. Remember I said I was making v=theta to make it easier to read.