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General Antiderivative

  • Thread starter frumdogg
  • Start date
18
0
1. Homework Statement

Hi everyone. My Calc 1 final exam is tomorrow and due to some weather related issues we were not able to cover all material for this quarter. With that said, my professor gave us a take home quiz on material that was unable to be covered. I have done by best, but I am getting hung up on an antiderivative problem.

Find the general antiderivative of:


f'(x) = 1-2x-4/[tex]\sqrt{x}[/tex]+5/x-8/(1+x^2)+9/x^4

2. Homework Equations

Now due to having virtually no time to learn about antiderivatives (we lost a whole week due to a blizzard and instructor illness) I am really unsure where to go. Do I need to rewrite the problem on one line and then find the opposite of the derivative?

Thanks!
 

Answers and Replies

malawi_glenn
Science Advisor
Homework Helper
4,782
22
well yes, what function, if you differentiate it, will give that expression as its derivative?
 
18
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So basically the definition of an antiderivative is the oppositive of the derivative?

Wait, I did not just type that last sentence. =)
 
1,341
3
Do you mean:

[tex]f'(x) = 1-2x-4\sqrt{x}+\frac{5}{x}-\frac{8}{1+x^2}+\frac{9}{x^4}[/tex] ?

Basically when you're asked to find the anti-derivative you're trying to find the function f(x), which has this derivative f'(x), which is given.
 
Last edited:
18
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1- 2x -4[tex]\sqrt{x}[/tex]+[tex]\frac{5}{x}[/tex]-[tex]\frac{8}{1+x^{2}}[/tex]+[tex]\frac{9}{x^{4}}[/tex]
 
malawi_glenn
Science Advisor
Homework Helper
4,782
22
yes!

But you often write it given f(x), find its primitive function F(x): F' = f
 
18
0
That is sort of how I figured that it would work. My derivative skills are giving me the most trouble at this point.
 
138
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Do you have a textbook? First, read the section on anti-derivatives.

Hint: separate each quantity
f '(x) = 1, then f(x) = x + C.
f '(x) = 2x, then f(x) = ?
f '(x) = 9x^-4, then f(x) = ?
.
.
.
and so on.
 
18
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f '(x) = 2x, then f(x) = x^2
f '(x) = 9x^-4, then f(x) = 9x^-3/-3
correct?
 
458
0
The general antiderivative will have a + C at the end.
 
1,341
3
f '(x) = 2x, then f(x) = x^2
f '(x) = 9x^-4, then f(x) = 9x^-3/-3
correct?
So far, so good

But Snazzy is right you need to add a +C (constant of integration) term to the anti derivative, because if f(x) had a constant value some where in it, like f(x) = 2x+5, f'(x) = 2 -- so when we integrate f'(x) we need to account for the 5. We don't necessairly know it's a 5 so that's why we add the +C
 
Last edited:
18
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It's so easy to forget the C at the end.
 

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