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 Cfor the...

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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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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