How do you anti-derive x(2-x)^2?

Question

anonymous · Answer

you have to take the integral

anonymous · Answer

Would you mind showing me step by step?

anonymous · Answer

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 $$\int\limits 4x-4x ^{2}+x^3 dx$$

anonymous · Answer

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...

anonymous · Answer

correcto mundo

anonymous · Answer

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

anonymous · Answer

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.

anonymous · Answer

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.

anonymous · Answer

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.