Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Differentiation takes a function and produces its rate of change. Antidifferentiation reverses that: given f(x), you look for F(x) with F'(x) = f(x). Because the derivative of any constant is zero, if F(x) works then so does F(x) + 5, F(x) - 100, or F(x) plus anything constant. So the indefinite integral, written as the integral sign with f(x) dx, is not one function but a whole family: F(x) + C. That plus C is not decoration. It is the honest admission that differentiation throws away constant information, and antidifferentiation can never get it back unless you are given extra information, like a point the function passes through.

To build the basic integral table, run each derivative rule in reverse. Since d/dx[xⁿ] = n x^n-1, the power rule backwards says the integral of xⁿ dx is xⁿ⁺¹/(n+1) + C, for n not equal to -1. That exception matters: since d/dx[ln(x)] = 1/x, the integral of 1/x dx is ln|x| + C. Since d/dx[e^x] = e^x, e^x integrates to itself. Each term in a sum integrates separately, and constants multiplied through just ride along, so a sum like 4x³ - 6/x + 2e^x breaks into three familiar pieces.

This topic hands you the raw mechanics that every later technique in the unit depends on: substitution, integration by parts, and partial fractions are all ways of disguising a harder integral until it matches one of these basic forms. It also connects straight back to the Fundamental Theorem of Calculus, where evaluating a definite integral means finding any antiderivative F and computing F(b) - F(a), a step where the plus C conveniently cancels out.