Integrals

Maths · Class 12

Simulation · Maths · Class 12

One derivative, a whole family

"One derivative, a whole family" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Antiderivatives and the constant C" in Integrals. Each curve y = x² + C has derivative 2x, so each is an antiderivative of 2x. Changing C only lifts or drops the curve. The dashed line carries a short tangent on every curve: however far you drag, those tangents stay parallel. ∫2x dx = x² + C: a family of vertical shifts with parallel tangents above every x. A single point on the curve fixes C. JEE Main asked 73 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Antiderivatives and the constant C in Integrals. Change the values and watch what happens.

One derivative, a whole familyMaths · Class 12

In JEE Main

42 shifts analysed

73 questions from Integrals across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §7.1–7.2

  • 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.

Questions

3 answers

What does the "One derivative, a whole family" simulation show?
Each curve y = x² + C has derivative 2x, so each is an antiderivative of 2x. Changing C only lifts or drops the curve. The dashed line carries a short tangent on every curve: however far you drag, those tangents stay parallel.
Does JEE Main ask questions from Integrals?
Yes. JEE Main asked 73 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Antiderivatives and the constant C" in Integrals (Class 12 Maths).

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