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1. Jan 31, 2017

jlmccart03

1. The problem statement, all variables and given/known data
Integrate x2(2+x3)4dx.

2. Relevant equations
No equations besides knowing that the integral of xpower is 1/power+1 * xpower + 1

3. The attempt at a solution
So I have the answer to the integral by hand as (2+x3)5)/15 + C.
When I go to Wolfram Alpha it gives x15/15 + 2x12/3 + 8x9/3 + 16x6/3 + 16x3/3 + C

I really truly have no idea how these two are the same. I tried multiple types of manipulation to my answer, but I am completely lost on where basically every factor comes from besides the first x15/15.

Any help will be appreciated!!!

2. Jan 31, 2017

Math_QED

They expanded $(2+x^3)^5$ as $x^{15} + 10 x^{12} + 40 x^9 + 80 x^6 + 80 x^3 + 32$ (this can be found using Newton's binomium). Thus, after dividing both sides with 15, we get:

$\frac{(2+x^3)^5}{15} = x^{15} /15 + 2x^{12}/3 + 8x^9/3 + 16x^6/3 + 16x^3/3 + 32/15$

However, the constant in the primitive function does not matter (as we write $+ c$ anyway), so we can drop the $32/15$ safely.

3. Jan 31, 2017

jlmccart03

Ok, So they simply expanded the numerator using a thing called Newton's binomium? I will do some research on that, but I do not think we have ever learned what Newton's Binomium is. Thanks for explaining how this worked. I was completely lost on how it worked, but now it seemed relatively simple.

4. Jan 31, 2017

Math_QED

5. Jan 31, 2017

jlmccart03

OHHHHHHH that is what that is called. Ok, so I have done that before. Totally did not think of that as a solution. Thanks for the link, totally forgot that I could use that method.