Anti-Derivatives with Substitution: Solving x^2(1-x)^8dx

[A_Munk3y]

Homework Statement

Well, i didn't know how to do anti-derivatives on this forum so i just did it on paint :)
Anyways, here is the problem and solution i tried.
Let me know if i did anything wrong, or if i even did anything right...
Thanks a lot!

i have to find the anti derivative of : x²(1-x)⁸dx

The Attempt at a Solution

 

[Juanriq]

Hey a Munk3y, it looks good exceot for the multiplication of $(1-u)^2u^8$. try expanding the squared term first...

 

[A_Munk3y]

yea, i thought that was wrong... I've always sucked at doing this..

so (1-u)²=u²-2u+1
then, u⁸(u²-2u+1) right?

would it be u⁸-2u⁹+u¹⁰

 

[Juanriq]

Thats absolutely right. Alot of people don't even see making the substitution for x afterwards, they get stuck after the u-substitution.

 

[A_Munk3y]

ok thanks , but now I'm a little lost on what to do now with this...
i tried to change it since i have a new equation but i think I'm doing something wrong.
either I'm doing the signs wrong, or something else, but it just doesn't seem right

here is what i got

 

[Juanriq]

No that is fine! You can simplify the -2/10 to -1/5, and rewrite the terms in order of decreasing exponents, but only the evilest of teachers would expect you to expand those terms!

 

[Juanriq]

Also, I commend you on using paint. That takes dedication!

 

[A_Munk3y]

heh... yea, paint is a pain , but it's still easier than trying to figure out how to do it on this forum
and my teacher told us we don't have to simplify, so I'm good :Done last thing to be sure... :P I'm right in that only the 2/10(1-x)¹⁰ is negative? I thought i had messed that one up :shy:

 

[Juanriq]

Actually all the signs are bacwards, you lost the negative sign right after your first equal sign.

 

[Juanriq]

You get du = -dx. This implies that -du = dx as well. The intermediate step in your solution would be $- \int u^8 - 2u^9 + u^{10} du = \int - u^8 + 2u^9 - u^{10} du$ and you would break up the integrals from there. LaTex isn't so bad to learn! I guarantee that it takes less time than paint =)

 

[A_Munk3y]

yea, i thought i had done something wrong
so is this right now?

 

[Juanriq]

You got it!

 

[A_Munk3y]

YEA! :)
thanks so much for the help!
I reaaaaaaaally appreciate it.