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

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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**

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**