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Given primitive F = (x^2-1)/(sqrt x), find the function f?

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ruals | Salutatorian

Posted September 13, 2013 at 2:19 PM via web

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Given primitive F = (x^2-1)/(sqrt x), find the function f?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 13, 2013 at 2:44 PM (Answer #1)

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You need to remember that the primitive integral is a differentiable function whose derivative is `f(x)` , such that:

`F'(x) = f(x)`

Hence, you need to differentiate the function `F(x)` with respect to `x` , using the quotient rule, such that:

`F'(x) = ((x^2-1)'(sqrt x) - (x^2-1)(sqrt x)')/((sqrt x)^2)`

`F'(x) = (2xsqrt x - (x^2-1)/(2sqrtx))/x`

`F'(x) = (4x*sqrt x*sqrt x - x^2 + 1)/(2xsqrt x)`

`F'(x) = (4x^2 - x^2 + 1)/(2xsqrt x)`

`F'(x) = (3x^2 + 1)/(2xsqrt x)`

Rationalizing yields:

`F'(x) = (sqrt x*(3x^2 + 1))/(2x^2) = f(x)`

Hence, evaluating the function `f(x)` , under the given conditions, yields `f(x) = (sqrt x*(3x^2 + 1))/(2x^2).`

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