Find the equation of the function f(x) if f'(x) = (x^3 - x) / 4x^4.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have f'(x) = (x^3 - x) / 4x^4

To find f(x), we need to integrate f'(x)

Int [ (x^3 - x) / 4x^4]

=> Int [ x^3/4x^4 - x/4x^4]

=> (1/4)*Int[1/x dx] - (1/4)*Int[1/x^3 dx]

=> (1/4)*ln |x| - (1/4)*x^-2 / (-2)

=> (1/4)*ln |x| + (1/8)*x^-2

We get f(x) = (1/4)*ln |x| + (1/8)*x^-2 + C

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given the derivative f'(x) = (x^3 - x)/4x^4

Let us simplify.

==> f'(x) = (x^3/4x^4) - x/4x^4

==> f'(x) = (1/4x) - 1/4x^3

==> f'(x) =(1/4)[ 1/x - x^-3)

Now we will integrate.

==> f(x) = Int f'(x) = (1/4) * Int ( 1/x - x^-3) dx

              = (1/4) [ lnx - x^-2/-2 ] + C

             = (1/4) l lnx + 1/2x^2)  + C

Then the function f(x) is given by : f(x) = (1/4) *[ ln x + 1/2x^2] + C

 

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