In case this looks familiar, yes, it is a repost. I really need help.
Find the anti-derivitive of
f(x)= x^3(x-2)^2.
Multiple choice gives me the options:
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 got as far as factoring the original equation into x^5 - 4x^4 + 4x^2. I was told I need to integrate the function. I'm not sure how to do that. My book is NOT clear or helpful, and hasn't been for this whole calculus course.
Please help me understand how my equation can possibly be one of the equations in the A-B-C-D options.
Thank you and God bless you for taking the time to read this, let alone help!
Yes, you need to integrate the function.
I am sure antiderivatives and integration are in your text.
first multiply it out
x^3(x-2)^2 = x^3 (x^2 - 4 x + 4)
= x^5 -4x^4 + 4 x^3 (note that last x^3)
now in general the integral of x^n dx is (1/n+1)x^(n+1)
so for example the integral of x^2dx is
(1/3)x^3 + a constant C
so here we have
(1/6)x^6 -(4/5)x^5 + x^4 + C
Which is answer (a)
try
http://archives.math.utk.edu/visual.calculus/4/antider.1/
Be sure to follow the tutorial links at the bottom left of that link.
Ah, thank you so much. Those are very helpful. Yes, anti-derivatives and integrals are in my text, but it is explained in one paragraph. Thanks again!
Now do you see what I meant about switching names in my earlier post
http://www.jiskha.com/display.cgi?id=1335474065
Both Damon and I spent time answering the same question.
I apologize. I didn't see the earlier post, but it won't happen again. Sorry to the both of you.
To find the antiderivative of the function f(x) = x^3(x-2)^2, you can use the power rule for integration. The power rule states that when integrating a function of the form x^n, where n is any real number except -1, the antiderivative is (1/(n+1)) * x^(n+1).
To integrate the given function, you can expand the expression inside the parentheses using the binomial theorem. First, distribute the x^3 to both terms within the parentheses:
f(x) = x^3 * (x-2)^2
= x^3 * (x^2 - 4x + 4)
Expanding further, you get:
f(x) = x^5 - 4x^4 + 4x^3
Now you can integrate each term separately using the power rule.
∫x^5 dx = (1/6) * x^6
∫-4x^4 dx = (-4/5) * x^5
∫4x^3 dx = (4/4) * x^4 = x^4
Adding these integrals together, you get:
∫f(x) dx = (1/6)x^6 - (4/5)x^5 + x^4 + C
So, the antiderivative of f(x) is given by option A: (1/6)x^6 - (4/5)x^5 + x^4 + C.
It is important to note that when integrating, you should always include the constant of integration (C) because the operation of integration is not unique. The constant accounts for the fact that there could be an infinite family of functions whose derivatives would match the given function.