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

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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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By definition, f(x) could be determined evaluating the indefinite integral of f'(x)

Int (236x^5+3x^2)dx

We'll apply the additive property of integrals:

Int (36x^5+3x^2)dx = Int (36x^5)dx + Int (3x^2)dx

We'll re-write the sum of integrals, taking out the constants:

Int (36x^5+3x^2)dx = 36Int x^5 dx + 3Int x^2 dx

Int (36x^5+3x^2)dx = 36*x^6/6 + 3*x^3/3

We'll simplify and we'll get:

Int (36x^5+3x^2)dx = 6x^6 + x^3 + C

The function f(x) is: f(x) = 6x^6 + x^3 + C

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Basically, what you are trying to find is the antiderivate of the function.

To find the antiderivative, you will have to solve using integral rules.

Antiderivate is f(x) = 36x^6 / 6 + 3x^2 / 3