How Can Trigonometric Substitution Simplify the Integral of x²/√(1-x²)?

[kathrynag]

Homework Statement

$$\int^{0}_{1}\frac{x^{2}}{\sqrt{1-x^{2}}}$$

Homework Equations

The Attempt at a Solution

Let x=sintheta
dx=cos theta
$$\int^{0}_{1}\sin^{2}$$
Now I get stuck

 

[sutupidmath]

well, i believe

$$sin^2x=\frac{1-cos2x}{2}$$ would help.

 

[Mark44]

You don't show dx in your original integral, but it should be there. You need to replace it and the square root in the denominator, using your substitution.

In your substitution, dx = cos(theta) d(theta).

 

[Dick]

There's a trig formula that let's you express sin(theta)^2 in terms of cos(2*theta). Can you find it?

 

[kathrynag]

well, i believe

$$sin^2x=\frac{1-cos2x}{2}$$ would help.

so then i would make u=2x and du=2dx

 

[sutupidmath]

so then i would make u=2x and du=2dx

well first you would break it into 1/2-1/2 cos(2x) then if you want you can make that substituion, that would work.

 

[kathrynag]

well first you would break it into 1/2-1/2 cos(2x) then if you want you can make that substituion, that would work.

Ok, well that's what I meant.

 

[sutupidmath]

Ok, well that's what I meant.