Determine the function f(x) if f'(x)=15x^14+36x^5-2x^3?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The derivative of f(x) is given, f'(x) = 15x^14+36x^5-2x^3

To find f(x) we integrate f'(x)

Int[f'(x) dx]

=> Int [ 15x^14+36x^5-2x^3 dx]

=> 15*x^15 / 15 + 36*x^6 / 6 - 2*x^4 / 4

=> x^15 + 6x^6 - (x^4)/2 + C

The function f(x) = x^15 + 6x^6 - (x^4)/2 + C

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the primitive of the original function, we'll have to determine te indefinite integral of the expression of derivative.

We'll determine the indefinite integral of

f'(x)=15x^14+36x^5-2x^3

Int f'(x)dx = f(x) + C

Int (15x^14+36x^5-2x^3)dx

We'll apply the property of the indefinite integral, to be additive:

Int (15x^14+36x^5-2x^3)dx = Int (15x^14)dx + Int (36x^5)dx - Int (2x^3)dx

Int (15x^14)dx = 15*x^(14+1)/(14+1) + C

Int (15x^14)dx = 15x^15/15 + C

Int (15x^14)dx = x^15 + C (1)

Int (36x^5)dx = 36*x^(5+1)/(5+1) + C

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

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

Int 2x^3dx = 2*x^4/4 + C

Int 2xdx = x^4/2 + C (3)

We'll add: (1)+(2)-(3)

Int (15x^14+36x^5-2x^3)dx = x^15 + 6x^6 - x^4/2 + C

So, the function is:

f(x) = x^15 + 6x^6 - x^4/2 + C

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