find the antiderivative of f(x) = x^3(x-2)^2
I factored this out and got
x^5-4x^4+4x^2
I'm supposed to integrate this. I don't know how. Someone help me please?
BTW, multiple choice options are:
A
B
C
D
Sorry, let me fix that. Multiple Choice options are:
A: 1/6 x^6 - 4/5 X^5 + x^4 + C
B: 1/6 x^6 - 4/5 X^5 + 1/4 x^4 + C
C: x^6 - X^5 + x^4 + C
D: x^5 - 4x^4 + 4x^3 + C
I assume this is a continuation of
http://www.jiskha.com/display.cgi?id=1335468769
Please pick a "nickname" and stick to it, that way we can follow your posts easier.
You had asked for the anti-derivative, and I had replied with "integrate"
That is the more "mathematical" terminology.
To "integrate" something is the same as finding the "anti-derivative"
e.g.
the derivative of 6x^3 + 5x - 2 is 18x^2 + 5
and the integral or the anti-derivative of 18x^2 + 5 is 6x^3 + 5x
btw, I had a typo in my expanded form, should have been
x^5 - 4x^4 + 4x^3 , I am surprised you didn't check that
so the anti-derivative or the integral of
x^5 - 4x^2 + 4x^3 is
(1/6)x^6 - (4/5)x^5 + x^4 + C , where C is a constant
clearly choice a)
(differentiate choice a) and see what you get, to verify the answer)
To find the antiderivative of the function f(x) = x^3(x-2)^2, you can integrate each term separately using the power rule for integration.
Here are the steps to find the antiderivative:
1. Expand the expression: x^3(x-2)^2 = x^5 - 4x^4 + 4x^2.
2. Integrate each term separately:
- ∫x^5 dx = (1/6)x^6 + C1, where C1 is the constant of integration.
- ∫-4x^4 dx = - (4/5)x^5 + C2, where C2 is another constant of integration.
- ∫4x^2 dx = (4/3)x^3 + C3, where C3 is another constant of integration.
3. Combine the antiderivatives of each term to get the final antiderivative:
∫f(x) dx = (1/6)x^6 + C1 - (4/5)x^5 + C2 + (4/3)x^3 + C3.
Now, you have the expression for the antiderivative of f(x).
Regarding the multiple-choice options, you haven't provided them in your question, so it's not possible to determine the correct choice (A, B, C, or D) without further information.