Integration by Substitution

[bross7]

I'm stuck on how to advance further on this problem and if anyone can point my in the right direction I would be greatly appreciative.

\int\frac{dx}{\sqrt{x(1-x)}}

The integral has to be solved using substitution, but we are required to use
u=\sqrt{x}

From this:
du=\frac{dx}{2\sqrt{x}}

But I am stuck on how to convert the remaining portion of the function in terms of du.
\int\frac{dx}{u\sqrt{1-x}}

[ShawnD]

I gave it a try and couldn't get anywhere with it. Maple says the answer is arcsin(2x-1).

Is that exactly how the question was given?

[Justin Lazear]

u = \sqrt{x}

so

u^2 = x

and

2du = \frac{dx}{\sqrt{x}}

First use the third equation, then use the second equation to get rid of any other instances of x that're left.

And Shawn is not correct in his solution.

--J

[vladimir69]

Shaun's solution looks good to me, what do you propose the actual answer is Justin?

[spacetime]

Complete the square within the square root in the denominator and the apply the result

\int\frac{dx}{\sqrt{a^2-x^2}} = arcsin\frac{x}{a}

spacetime
www.geocities.com/physics_all

[Justin Lazear]

\int\frac{dx}{\sqrt{x(1-x)}} = 2 \arcsin{\left(\sqrt{x}\right)}

Differentiate it and you'll get the integrand.

The derivative of arcsin(2x-1) is \frac{2}{\sqrt{4x^2 - 4x + 2}}.

--J