Lesson 1 of 13 · 11 min
Antiderivatives and the constant C
NCERT §7.1–7.2
The pump on the school roof has a meter that reads 10 litres per minute at switch-on and climbs by 2 litres per minute every minute. The caretaker wants the amount of water in the tank after 10 minutes, and the meter alone cannot tell him.
The story this chapter follows: The rooftop water tank
The lesson in notes
In short
Differentiation takes a function and returns its rate of change. Integration asks the reverse question: which function F has the given f as its derivative? Any such F is an antiderivative (or primitive) of f.
Antiderivatives are never unique. The derivative of a constant is zero, so if F′ = f then (F + C)′ = f for every real C. On an interval, any two antiderivatives of the same f differ only by a constant.
The whole family is the indefinite integral: ∫f(x) dx = F(x) + C. Here f(x) is the integrand, x the variable of integration and C the constant of integration.
Geometrically the graphs of F(x) + C are copies of one curve slid up or down. Above any chosen x they all have the same slope, so their tangents there are parallel.
One known point on the curve fixes C. Example: if F′(x) = 4x³ − 6 and F(0) = 3, then F(x) = x⁴ − 6x + C and F(0) = 3 gives C = 3, so F(x) = x⁴ − 6x + 3.
Differentiation and integration undo each other up to a constant: the derivative of ∫f(x) dx is f(x), while ∫f′(x) dx is f(x) + C.
A derivative has a meaning at a single point. An indefinite integral does not: it describes a family of functions over an interval.