What is f(x) if f'(x)=25x^4+2e^2x ?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have f'(x) = 25x^4 + 2e^2x

f(x) = Int [ f'(x) dx]

=> f(x) = Int[ 25x^4 + 2e^2x dx]

=> f(x) = Int [ 25 x^4 dx] + Int [ 2e^2x dx]

=> f(x) = 25* x^5 / 5 + 2*e^2x/2 + C

=> f(x) = 5x^5 + e^2x + C

The required integral is 5x^5 + e^2x + C

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To calculate a function, when knowing it's derivative, we'll have to integrate the expression of derivative.

We'll determine the indefinite integral of f'(x)= 25x^4+2e^2x.

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

Int (25x^4+2e^2x)dx

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

Int (25x^4+2e^2x)dx  =Int (25x^4)dx + Int (2e^2x)dx 

Int (25x^4)dx = 25*x^(4+1)/(4+1) + C

Int (25x^4)dx = 25x^5/5 + C

Int (25x^4)dx = 5x^5 + C (1)

Int 2e^2xdx = 2*e^2x/2 + C

Int 2e^2xdx = e^2x + C (2)

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

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

So, the function is:

f(x) = 5x^5 + e^2x + C

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