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Class 12 Maths Notes

Integrals Class 12 Notes

Complete, exam-ready notes on integrals: the idea of integration as antiderivative, indefinite and definite integrals, substitution, integration by parts and partial fractions, and the fundamental theorem of calculus — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Integration reverses differentiation and measures exact area under a curve, using substitution, integration by parts and partial fractions plus the fundamental theorem of calculus.

Indefinite Integrals and Standard Forms

Antiderivative

An indefinite integral ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C is the family of functions whose derivative is f(x); C is the constant of integration.

  • Power rule:
  • ∫xn dx=xn+1n+1+C (n≠−1)\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\ (n \neq -1)
  • .
  • Standard forms:
  • ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx = \ln|x| + C
  • ,
  • ∫ex dx=ex+C\int e^x\,dx = e^x + C
  • ,
  • ∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx = -\cos x + C
  • .
  • Linearity:
  • ∫(af+bg) dx=a∫f dx+b∫g dx\int (af + bg)\,dx = a\int f\,dx + b\int g\,dx
  • .
∫xn dx=xn+1n+1+C,∫dxx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1}+C,\quad \int \frac{dx}{x} = \ln|x|+C
Power and reciprocal rules

Substitution and Partial Fractions

Substitution

Set u=g(x)u = g(x) so that du=g′(x) dxdu = g'(x)\,dx, converting a composite integrand into a standard form: ∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du.

  • Look for a function whose derivative is present: the signature of substitution.
  • Partial fractions break rational expressions into simpler pieces, e.g.
  • 1(x−1)(x−2)=Ax−1+Bx−2\frac{1}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}
  • .
  • Use substitution or completing the square when the denominator is quadratic.
∫f(g(x))g′(x) dx=∫f(u) du,u=g(x)\int f(g(x))g'(x)\,dx = \int f(u)\,du,\quad u = g(x)
Integration by substitution

Integration by Parts

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
Integration by parts
  • Choose u with the ILATE priority: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
  • Classic uses:
  • ∫x ex dx\int x\,e^x\,dx
  • ,
  • ∫ln⁡x dx\int \ln x\,dx
  • ,
  • ∫exsin⁡x dx\int e^x\sin x\,dx
  • (repeat by parts for cyclic integrals).
  • For ln x, take u = ln x and dv = dx.

ILATE decides u

Let u be the function that comes first in ILATE. Getting the order wrong is the most common error in integration by parts.

Definite Integrals and the Fundamental Theorem

Definite integral

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

The definite integral from a to b equals the change in the antiderivative; geometrically it is the signed area under the curve.

  • Additivity:
  • ∫ab+∫bc=∫ac\int_a^b + \int_b^c = \int_a^c
  • .
  • Symmetric-limit properties:
  • ∫0af(x) dx=∫0af(a−x) dx\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx
  • ,
  • ∫−aaf=2∫0af\int_{-a}^{a} f = 2\int_0^a f
  • for even f, and 0 for odd f.
  • King's property and
  • ∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx
  • simplify symmetric integrands.
∫abf(x) dx=F(b)−F(a),∫0af(x) dx=∫0af(a−x) dx\int_a^b f(x)\,dx = F(b)-F(a),\quad \int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx
Fundamental theorem and the symmetric property

Solved Examples

Example: Evaluate ∫01x ex dx\int_0^1 x\,e^x\,dx.

Solution: By parts with u=xu = x, dv=exdxdv = e^xdx: ∫x ex dx=xex−∫ex dx=ex(x−1)+C\int x\,e^x\,dx = xe^x - \int e^x\,dx = e^x(x-1) + C. So ∫01x ex dx=[ex(x−1)]01=0−(−1)=1\int_0^1 x\,e^x\,dx = [e^x(x-1)]_0^1 = 0 - (-1) = 1.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Power rule

∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1}+C

Substitution

∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du

Integration by parts

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du

Fundamental theorem

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b)-F(a)

Symmetric property

∫0af(x) dx=∫0af(a−x) dx\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx

Linearity

∫(af+bg) dx=a∫f dx+b∫g dx\int (af+bg)\,dx = a\int f\,dx + b\int g\,dx

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Always add the constant C for indefinite integrals.
  • ILATE order for integration by parts: Inverse, Log, Algebraic, Trig, Exponential.
  • Use ∫₀ᵃ f = ∫₀ᵃ f(a−x) when integrands are symmetric.
  • ∫₋ₐᵃ of an even function = 2∫₀ᵃ; an odd function gives 0.
  • Substitution works when the derivative of the inside function is present.
  • n ≠ −1 in the power rule; use ln|x| for 1/x.

FAQ

Common questions

Why do we add a constant of integration?

Because differentiating F(x) + C removes any constant. The derivative of x² + 5 and x² − 9 are both 2x, so the antiderivative is a whole family F(x) + C.

What is the difference between definite and indefinite integrals?

An indefinite integral is a family of antiderivatives with a constant C; a definite integral is a number, F(b) − F(a), equal to the signed area under the curve.

How do I choose u in integration by parts?

Use the ILATE order — Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential — and take u as the function appearing first.

What is the fundamental theorem of calculus?

It connects differentiation and integration: the definite integral of f from a to b equals the change in any antiderivative F, i.e. ∫ₐᵇ f(x)dx = F(b) − F(a).

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