# Integrate sqrt(x-x^2)

by autodidude
Tags: integrate, sqrtxx2
 P: 333 1. The problem statement, all variables and given/known data Integrate $$\sqrt{x-x^2}$$ The attempt I did a trig substitution, letting $$cos(\theta)=\frac{x}{sqrt(x)}$$ and after some manipulation ended up with $$-2\int \ |sin(\theta)cos(\theta)|sin(\theta)cos(\theta) d\theta$$ which I have no idea how to integrate. If I make a u-substitution and let u=cos(theta) rather than simplify to get the above, I get $$2\int \ u\sqrt{u^2-u^4}du$$ which I cant make any progress on either.
HW Helper
Thanks
P: 7,923
 Quote by autodidude $$-2\int \ |sin(\theta)cos(\theta)|sin(\theta)cos(\theta) d\theta$$
The original integral must be over a range in [0, 1]. This means you can restrict theta to [0, pi/2], allowing you to drop the modulus function, leaving sin2cos2. Can you solve it from there?
 HW Helper Sci Advisor Thanks P: 24,460 The more common way to do a problem like this is to complete the square inside the radical then substitute. I think it goes a bit easier that way.
P: 333

## Integrate sqrt(x-x^2)

@haruspex: Yeah, I tried that and when I got the incorrect answer, I went back and saw that I overlooked the fact that you need to insert the modulus wheen rooting a square. Will try again in case I made an error though.

@Dick: Thanks, I'll see where I can get with that.
 P: 572 Like Dick said. Look at it like this try to reformulate it so you get something like this: $$\int\sqrt{\frac{1}{4}-(x-}\frac{1}{2})^{2}dx$$ and substitute u : $$u=x-\frac{1}{2};dx=du$$ and see what you can get.
 P: 657 try factorizing out the x... then use a substitution sqrt x = something... simplifies things alot!

 Related Discussions Calculus & Beyond Homework 5 Calculus & Beyond Homework 7 Calculus & Beyond Homework 12 Calculus 4 Calculus 3