Given f'(x)=25x^4+6x determine f(x).

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neela | High School Teacher | (Level 3) Valedictorian

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f'(x) = 25x^4 +6x. To find f(x).

Therefore f(x) is the antiderivative of f'(x) or integral of (25x^4+6x) dx.

Therefore f(x) = Int f'(x) dx.

f(x) = Int (25x^4+6x) dx.

f(x) = Int 25x^4 dx +Int  6x dx.

f(x) = 25* (1/5) x^5 + 6* (1/2) x^2+ C.

 Therefore f(x) = 5x^5 +3x^2 + C.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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

Int (25x^4+6x)dx

We'll apply the additive property of integrals:

Int (25x^4+6x)dx = Int (25x^4)dx + Int (6x)dx

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

Int (25x^4+6x)dx = 25Int x^4 dx + 6Int x dx

Int (25x^4+6x)dx = 25*x^5/5 + 6*x^2/2

We'll simplify and we'll get:

Int (25x^4+6x)dx = 5x^5 + 3x^2 + C

The function f(x) is: f(x) = 5x^5 + 3x^2 + C

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