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!