Solving an Integral Involving sqrt(1-x^2) and cos(Theta)^2

  • Thread starter Thread starter Sheneron
  • Start date Start date
  • Tags Tags
    Integral
Sheneron
Messages
360
Reaction score
0

Homework Statement



Integral of sqrt(1-x^2) dx

I subbed in sin(Theta) for x, and did my work and got the integral down to: cos(Theta)^2 dTheta.

The problem I am having is integrating cos(Theta)^2. I could use a half angle formula, but the problem is I have to rewrite theta in terms of x, and if I use have angle I end up with Cos(2Theta), which I don't know how to convert back into x. Please help.
 
Physics news on Phys.org
Nevermind...

Found out the trig property I needed.

sin(2x) = 2sin(x)cos(x)
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Line Integral of circle in counterclockwise direction
Replies
5
Views
2K
Potential of a rotationally symmetric charge distribution
Replies
4
Views
1K
How should I show the following by using the signum function?
Replies
14
Views
2K
Calculating radius of gyration of plane figure about x-axis
Replies
5
Views
2K
Integrals that keep me up at night
Replies
22
Views
3K
How should I use the averaging approximation to find this?
Replies
3
Views
2K
Calculate surface integral on sphere
Replies
7
Views
2K
Center of mass of spherical shell inside of cone (Apostol Problem).
Replies
3
Views
2K
Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4
Replies
1
Views
2K
How to Integrate [1/(x^2 + 3)] dx?
Replies
5
Views
3K

Hot Threads

Recent Insights

Back
Top