Finding Antiderivatives and Indefinite Integrals

Derivatives destroy constants: the derivative of x² and the derivative of x² + 9 are both 2x. So when you work backwards from a derivative to find the original function, you can never know which constant was there originally. That's why every antiderivative is written with a +C tacked on, called the constant of integration. Without extra information, like a value of the function at a point, +C stays as a placeholder for every possible answer.

To build an antiderivative, run the power rule in reverse. The power rule for derivatives brings the exponent down and subtracts one from it. So reversing it means adding one to the exponent and dividing by that new exponent: the antiderivative of xⁿ is xⁿ⁺¹/(n+1) + C, as long as n is not -1. Apply this term by term to a polynomial. For 6x² - 4x + 5, antidifferentiate each piece separately: 6x² becomes 2x³, -4x becomes -2x², and the constant 5 becomes 5x. The general antiderivative is 2x³ - 2x² + 5x + C. If you're told a specific value like F(0) = 0, plug it in to solve for C and pin down one exact function out of that whole family.

This process is the reverse of the derivative rules from earlier units, and it's exactly the machinery the Fundamental Theorem of Calculus needs. To evaluate a definite integral, you first find a general antiderivative, then plug in the upper and lower bounds and subtract. Finding F(x) here is the setup step; evaluating F(b) - F(a) is the payoff that turns an infinite sum of slivers into one subtraction.