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