Find the most general antiderivative. (Use C for the constant of integration. Remember to use absolute values where appropriate.)
f '(x) = 4x^3 − 10x + 4
just use the power rule: ∫ x^n dx = 1/(n+1) x^(n+1)
So,
f(x) = x^4 - 5x^1 + 4x + C
To find the most general antiderivative of a function, we need to integrate each term separately and then add a constant of integration, represented by C.
Given f'(x) = 4x^3 - 10x + 4, we can integrate each term:
∫(4x^3) dx = x^4 + C1, where C1 is the constant of integration for the first term.
∫(-10x) dx = -5x^2 + C2, where C2 is the constant of integration for the second term.
∫(4) dx = 4x + C3, where C3 is the constant of integration for the third term.
Now, we add the antiderivatives of each term together:
f(x) = x^4 + C1 - 5x^2 + C2 + 4x + C3
To simplify the equation, we can combine the constants of integration into a single constant:
f(x) = x^4 - 5x^2 + 4x + C
So, the most general antiderivative of f'(x) = 4x^3 - 10x + 4 is f(x) = x^4 - 5x^2 + 4x + C, where C is the constant of integration.