## anonymous 5 years ago Integrate (x) (1-x)^(1/2) dx not sure where to even start....thanks

1. anonymous

try u = 1-x

2. anonymous

making x = 1-u

3. anonymous

$\int\limits\limits\limits_{?}^{?} x \sqrt{1-x} dx$

4. anonymous

giving $-\int(1-u)\sqrt(u)du=-\int u^{\frac{1}{2}}-u^{\frac{3}{2}}du$ etc

5. anonymous

using equation thing is not very intuitive is it? trying u = 1-x

6. anonymous

well no, but it is a standard trick.

7. anonymous

you have to deal with the annoying 1-x inside the radical somehow.

8. anonymous

so even though the derivative of 1-x would be -1 I just keep marching?

9. anonymous

yes. you have $u=1-x$ $du=-dx$ or $dx=-du$ and since $u=1-x$ we know $x=1-u$ to get $\int x sqrt{1-x}dx=-\int (1-u) \sqrt{u}du = \int u^{\frac{3}{2}}du - \int u^{\frac{1}{2}}du$

10. anonymous

i assume it is ok from there. take the antiderivative using the power rule and then replace u by 1-x

11. anonymous

- 2/3 (1-x)^ 3/2 + 2/5 (1-x)^5/2 +C

12. anonymous

that is it!