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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