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

Asked on by penarul

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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.`

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

nvoron23 | Student, College Freshman | (Level 1) eNoter

Posted on

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

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