indefinite integral

An indefinite integral is just a name and a notation for the whole family of antiderivatives of a function. Where the antiderivative idea says 'find a function whose rate is f,' the indefinite integral is the tidy way of writing down all the answers at once. Think of it as the recipe card: not one dish, but every dish that comes out tasting the same — all the functions that, when you take their derivative, give you back f.

We write it with the integral sign: integral f(x) dx = F(x) + C, read as 'the integral of f of x, dee x.' Here F is any one antiderivative of f, and the '+ C' carries along the family of every possible constant offset. The 'dx' marks which variable you are reversing the derivative with respect to. So integral 2x dx = x^2 + C, and integral cos(x) dx = sin(x) + C. The whole expression names a set of functions, not a number.

This is the crucial contrast with the definite integral, which carries limits like integral from a to b and evaluates to a single number. The indefinite integral has no limits and stays a function (family). The two share the integral sign because the Fundamental Theorem of Calculus links them: you compute a definite integral by first finding an indefinite integral. Do not let the shared symbol fool you into thinking an indefinite integral is a quantity — it is a function with a free constant attached.