How to find the function if the derivative is given? Given f'(x)=36x^5+3x^2 determine f(x).

Expert Answers

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The problem provides the derivative of the function, hence, you need to evaluate the function using anti-derivatives, such that:

`int f'(x) = f(x) => int (36x^5+3x^2)dx = f(x)`

You need to use the linearity of integral, hence, you need to split the integral in two simpler integrals, such that:

`int (36x^5) dx + int (3x^2)dx = 36 int x^5 dx + 3 int x^2 dx`

`int (36x^5) dx + int (3x^2)dx = 36 x^6/6 + 3 x^3/3 + c`

Reducing duplicate factors yields:

`int (36x^5) dx + int (3x^2)dx = 6x^6 + x^3 + c`

Hence, evaluating the function f(x) using anti-derivatives, yields `f(x) = x^3(6x^3 + 1) + c.`

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