Lesson 3 of 13 · 6 min
Properties of the indefinite integral
NCERT §7.2.1
Suppose the tank also collects water from a rain pipe at a steady 6 litres per minute while the pump runs. How much arrives from the two together?
The lesson in notes
In short
The derivative of ∫f(x) dx is f(x), and ∫f′(x) dx = f(x) + C.
Two indefinite integrals with the same derivative describe the same family of curves; they may look different but differ by a constant.
Integrals add term by term: ∫(f + g) dx = ∫f dx + ∫g dx.
A constant factor comes out: ∫k f(x) dx = k∫f(x) dx for any real k.
Combining the last two, ∫[k₁f₁ + k₂f₂ + … + kₙfₙ] dx = k₁∫f₁ dx + k₂∫f₂ dx + … + kₙ∫fₙ dx: integrate term by term.
Worked example: ∫(6x² − 4/x + 2eˣ) dx = 2x³ − 4 log|x| + 2eˣ + C. One constant C is enough for the whole answer.
There is no product rule for integrals: ∫f g dx is not ∫f dx × ∫g dx. Products need substitution, identities or integration by parts.