Integrals

Board: CBSE Class: 12

Chapter Overview

Integration is the reverse of differentiation. Indefinite integrals represent the family of functions whose derivative is the integrand, with a constant of integration. Methods include substitution, integration by parts, partial fractions, and special integrals. The definite integral is the limit of a sum representing area under the curve. The Fundamental Theorem of Calculus connects definite and indefinite integrals.

Topics Covered

Indefinite Integrals
Basic Integration Formulas
Integration by Substitution
Integration by Parts
Partial Fractions
Special Integrals
Definite Integrals
Limit of a Sum
Fundamental Theorem
Properties of Definite Integrals

Key Formulas

int xⁿ dx = xⁿ⁺¹/(n+1) + C

int e^x dx = e^x + C

int u dv = uv - int v du

int_a^b f(x) dx = F(b) - F(a)

int_0^a f(x) dx = int_0^a f(a-x) dx

Real-World Applications

Applications: Area calculations, volume of solids, work in physics, probability distributions.

Study Tips

Tip: Learn standard integrals by heart

Tip: Choose substitution wisely

Tip: ILATE rule for integration by parts