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Find the most general antiderivative of the function. (Check your answer by...

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fahriye23 | Student, Undergraduate | (Level 1) eNoter

Posted November 20, 2012 at 10:57 PM via web

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Find the most general antiderivative of the function. (Check your answer by differentiation.

Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use ln |u| where appropriate.) 

f(x)= (2/5)-(6/x)

F(x)=___________________?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 21, 2012 at 7:27 AM (Answer #1)

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You need to find the antiderivative of the function f(x), hence, you need to integrate f(x) such that:

`int f(x) dx = F(x)  +c`

Substituting `2/5- 6/x`  for `f(x)`  yields:

`int (2/5 - 6/x)dx `

Using the property of linearity of integrals yields:

`int (2/5 - 6/x)dx = int (2/5) dx- int (6/x)dx `

You need to take out the constants such that:

`int (2/5 - 6/x)dx = (2/5) int dx - 6 int 1/x dx`

`int (2/5 - 6/x)dx = (2/5)x - 6 ln|x| + c`

Hence, evaluating  `F(x)`  yields `F(x) = (2/5) x - 6 ln|x| + c.`

You need to differentiate the function `F(x)`  with respect to x to check if it yields `f(x)`  such that:

`F'(x) = (2/5)*x' - 6*(ln|x|)' + c'`

`F'(x) = (2/5)*1 - 6*1/x + 0`

`F'(x) = 2/5 - 6/x = f(x)`

Hence, evaluating the antiderivative of the given function f(x) yields `F(x) = (2/5) x - 6 ln|x| + c` .

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