## anonymous 5 years ago How do you anti-derive x(2-x)^2?

1. anonymous

you have to take the integral

2. anonymous

Would you mind showing me step by step?

3. anonymous

I'm a little rough at this but I can try to help. I would go ahead and multiply through before calculating the integral. So you would have $x(2-x)(2-x)$ then you would foil and multiply through by x. You end up with an integral that looks like this $\int\limits 4x-4x ^{2}+x^3 dx$

4. anonymous

Check my work... Like I said I'm a little rough at this. I believe you use a basic integration formula for each of those numbers...

5. anonymous

correcto mundo

6. anonymous

$\int\limits 4x dx = 4(1/2)x ^{2}$

7. anonymous

You can make it much simpler by letting u=2-x and du=-1*dx. Then, you get -∫u^2*(2-u) du which expands to -∫2u^2-u^3 du = -[2u^3/3 - u^4/4] + c = -2(2-x)^2/3 + (2-x)^4/4 + c, if I'm not mistaken. MtHaleyGirl's solution is correct, but just seems like a lot more tedious multiplication than is necessary.

8. anonymous

I DO take the long way... Scared of substitution but I just need a little practice. Good tip. Maybe it will make MY life easier.

9. anonymous

Yesss, substitution makes life so much easier later on, once you get confident with it. It's an absolute must for a lot, if not all, of advanced techniques of integration.