Integrating \sqrt{x-x^2} using Trig Substitution

[autodidude]

Homework Statement

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 can't make any progress on either.

 

[haruspex]

$$-2\int \ |sin(\theta)cos(\theta)|sin(\theta)cos(\theta) d\theta$$

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?

 

[Dick]

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.

 

[autodidude]

@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.

 

[mtayab1994]

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.

 

[unscientific]

try factorizing out the x... then use a substitution sqrt x = something... simplifies things a lot!