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Integrals

NCERT Class 12 Mathematics Chapter 7

Integrals for Class 12 Maths and JEE: anti derivatives, substitution, partial fractions, integration by parts, definite integrals and their properties.

9 topics Updated 8 Nov 2026

For: Class 12 and JEE

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part II (Class 12), Chapter 7 “Integrals”, reprint 2026-27.

Open the official NCERT chapter (PDF, 5.7 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Integration as anti differentiation

    Indefinite integrals for Class 12 Maths and JEE: anti derivatives, the constant C, standard integration formulas, log |x|, properties, solved examples.

    NCERT Section 7.2 13 min read 4 solved examples 11 practice questions

  2. Methods of integration: substitution and identities

    Integration by substitution for Class 12 Maths and JEE: choosing t, integrals of tan x, cot x, sec x, cosec x, trigonometric identities, solved examples.

    NCERT Section 7.3 15 min read 5 solved examples 12 practice questions

  3. Integrals of some particular functions

    Integrals of particular functions for Class 12 Maths and JEE: 1/(x ± a), square roots, completing the square, linear numerators, solved examples.

    NCERT Section 7.4 15 min read 5 solved examples 12 practice questions

  4. Integration by partial fractions

    Integration by partial fractions for Class 12 Maths and JEE: improper fractions, linear, repeated and quadratic factors, finding A, B, C, solved examples.

    NCERT Section 7.5 14 min read 5 solved examples 12 practice questions

  5. Integration by parts

    Integration by parts for Class 12 Maths and JEE: the formula, ILATE for the first function, ∫e[f + f′] dx, square-root integrals, solved examples.

    NCERT Section 7.6 15 min read 5 solved examples 13 practice questions

  6. Definite integral

    Definite integral for Class 12 Maths and JEE: lower and upper limits, F(b) − F(a), signed area under a curve, the limit of a sum, solved examples.

    NCERT Section 7.7 10 min read 5 solved examples 10 practice questions

  7. Fundamental theorem of calculus

    Fundamental theorem of calculus for Class 12 Maths and JEE: first and second theorems, area function, F(b) − F(a), d/dx of an integral, solved examples.

    NCERT Section 7.8 13 min read 5 solved examples 10 practice questions

  8. Definite integrals by substitution

    Definite integrals by substitution for Class 12 Maths and JEE: changing the limits, limits that swap order, and NCERT Exercise 7.9 integrals solved.

    NCERT Section 7.9 10 min read 5 solved examples 10 practice questions

  9. Properties of definite integrals

    Properties of definite integrals for Class 12 Maths and JEE: P to P, King’s rule with a + b − x, even and odd functions, [0, 2a], solved examples.

    NCERT Section 7.10 13 min read 5 solved examples 14 practice questions

Key formulas in this chapter

Anti derivatives

  • ∫x dx = x/(n + 1) + C, n ≠ −1
  • ∫(1/x) dx = log |x| + C
  • ∫e dx = e + C; ∫a dx = a/log a, a > 0, a ≠ 1 + C

Substitution and identities

  • ∫f(g(x)) g′(x) dx = ∫f(t) dt, t = g(x)
  • ∫tan x dx = log |sec x| + C
  • ∫cot x dx = log |sin x| + C
  • ∫sec x dx = log |sec x + tan x| + C
  • ∫cosec x dx = log |cosec x − cot x| + C

Particular integrals

  • ∫dx/(x + a) = (1/a) tan (x/a) + C
  • ∫dx/√(a − x) = sin (x/a) + C
  • ∫dx/(x − a) = (1/(2a)) log |(x − a)/(x + a)| + C
  • ∫dx/√(x ± a) = log |x + √(x ± a)| + C

Partial fractions

  • A/(x − a) → A log |x − a|
  • B/(x − a) → −B/(x − a)
  • (px + q)/(x + 1) → (p/2) log (x + 1) + q tan x

Integration by parts

  • ∫f g dx = f ∫g dx − ∫(f′ ∫g dx) dx
  • ∫e[f(x) + f′(x)] dx = e f(x) + C
  • ∫log x dx = x log x − x + C
  • ∫√(a − x) dx = (x/2)√(a − x) + (a/2) sin (x/a) + C

Definite integral

  • ∫ f(x) dx from a to b = F(b) − F(a), for continuous f and F′ = f
  • a: lower limit; b: upper limit

Fundamental theorem

  • A(x) = ∫ f(t) dt from a to x ⇒ A′(x) = f(x)
  • ∫ f(x) dx from a to b = F(b) − F(a)
  • d/dx of ∫ f(t) dt from a to g(x) = f(g(x)) g′(x)

Substitution with limits

  • t = g(x): limits a, b → g(a), g(b)
  • ∫ f(g(x)) g′(x) dx from a to b = ∫ f(t) dt from g(a) to g(b)

Properties of definite integrals

  • ∫ f(x) dx = ∫ f(a + b − x) dx, both from a to b
  • For an existing integral: f even gives twice the integral from 0 to a; f odd gives 0 over [−a, a]
  • ∫ log sin x dx from 0 to π/2 = −(π/2) log 2

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