Integrals_Class12_Maths Power Point Presentation

Integrals - Class 12 Mathematics
Based on Latest CBSE Syllabus
Simple Explanation with Examples
and Important Questions

Introduction to Integrals
• Integration is the reverse process of
differentiation.
• It helps in finding the area under curves and
solving differential equations.

Indefinite Integrals
• An indefinite integral has no limits and
represents a family of functions.
• Example:
• ∫x dx = (x²/2) + C

Integration as Inverse of Differentiation
• If f'(x) = g(x), then ∫g(x) dx = f(x) + C.
• Example:
• Since d/dx (sin x) = cos x,
• ∫cos x dx = sin x + C

Geometrical Interpretation of Indefinite Integral
• It represents a family of curves obtained by
shifting the graph of an antiderivative
vertically.
• Used to find area under curves.

Properties of Indefinite Integrals
• 1. ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
• 2. ∫k*f(x) dx = k * ∫f(x) dx

Some Standard Integrals
• ∫1 dx = x + C
• ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
• ∫e^x dx = e^x + C
• ∫1/x dx = ln|x| + C

Methods of Integration
• 1. Substitution Method
• 2. Integration by Parts
• 3. Partial Fractions
• Example (Substitution):
• ∫2x * √(1 + x²) dx
• Let 1 + x² = t

Integration by Substitution
• Replace part of the integrand with a variable
to simplify.
• Example:
• ∫x * cos(x²) dx
• Let x² = t dx = dt/2x
⇒

Integration Using Partial Fractions
• Used for rational functions.
• Example:
• ∫(2x+3)/[(x+1)(x+2)] dx

Integration by Parts
• Formula: ∫u*v dx = u∫v dx − ∫(du/dx * ∫v dx) dx
• Use ILATE to choose u (Inverse, Log, Algebraic,
Trig, Expo).

Definite Integrals
• Has limits a and b. Represents exact area.
• Example:
• ∫₀¹ x² dx = [x³/3]₀¹ = (1/3) - 0 = 1/3

Fundamental Theorem of Calculus
• It links the concept of differentiation and
integration.
• If F is antiderivative of f, then:
• ∫_a^b f(x) dx = F(b) - F(a)

Properties of Definite Integrals
• 1. ∫_a^a f(x) dx = 0
• 2. ∫_a^b f(x) dx = -∫_b^a f(x) dx
• 3. ∫_a^b f(x) dx = ∫_a^b f(a + b - x) dx

Evaluation of Definite Integrals by Substitution
• Use substitution with limits adjustment.
• Example:
• ∫₀⁴ x / √(1 + x²) dx
• Let 1 + x² = t

Important Formulae of Integrals
• Standard Integrals:
• ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C, where n ≠ -1
• ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C
• ∫aˣ dx = aˣ / ln(a) + C, a > 0, a ≠ 1
• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C

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