NCERT Mathematics Class 12 - Chapter 7: Integrals - Notes

Learning Objectives

Understand indefinite integrals and integration techniques
Learn substitution, partial fractions, and integration by parts
Study definite integrals and the Fundamental Theorem of Calculus
Apply properties of definite integrals

Key Concepts

Indefinite Integrals

∫f(x)dx = F(x) + C where F'(x) = f(x). C is the constant of integration. Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1). ∫(1/x)dx = ln|x| + C. ∫e^xdx = e^x + C. ∫a^xdx = a^x/ln a + C. ∫sin x dx = -cos x + C. ∫cos x dx = sin x + C. ∫sec²x dx = tan x + C. ∫cosec²x dx = -cot x + C. ∫sec x tan x dx = sec x + C. ∫cosec x cot x dx = -cosec x + C.

Integration by Substitution

Integration by Partial Fractions

For rational functions P(x)/Q(x) where degree of P < degree of Q. Decompose into simpler fractions. Types: (1) Non-repeated linear: A/(x-a) + B/(x-b). (2) Repeated linear: A/(x-a) + B/(x-a)². (3) Irreducible quadratic: (Ax+B)/(x²+bx+c). Equate numerators and solve for constants.

Integration by Parts

∫u dv = uv - ∫v du. Choice of u (ILATE rule): Inverse trig → Logarithmic → Algebraic → Trig → Exponential. Choose u as the function that comes first in ILATE.

Special form: ∫e^x[f(x) + f'(x)]dx = e^xf(x) + C (very useful shortcut for JEE).

Important Integral Forms

∫dx/√(a²-x²) = sin^-1(x/a) + C. ∫dx/(a²+x²) = (1/a)tan^-1(x/a) + C. ∫dx/(x²-a²) = (1/2a)ln|(x-a)/(x+a)| + C. ∫dx/√(x²+a²) = ln|x + √(x²+a²)| + C. ∫√(a²-x²)dx = (x/2)√(a²-x²) + (a²/2)sin^-1(x/a) + C.

Definite Integrals

Fundamental Theorem: ∫_a^b f(x)dx = F(b) - F(a) where F'(x) = f(x).

Properties: (1) ∫_a^b = -∫_b^a. (2) ∫_a^b = ∫_a^c + ∫_c^b. (3) ∫₀^a f(x)dx = ∫₀^a f(a-x)dx (King's rule). (4) ∫_-a^a f(x)dx = 2∫₀^a f(x)dx if f is even; = 0 if f is odd. (5) ∫₀^2a f(x)dx = 2∫₀^a f(x)dx if f(2a-x) = f(x); = 0 if f(2a-x) = -f(x).

Summary

Integration is the reverse of differentiation. Techniques include substitution, partial fractions, and integration by parts (ILATE rule). Definite integrals are evaluated using the Fundamental Theorem. Properties like King's rule and even/odd function properties simplify definite integrals significantly.

Important Terms

Antiderivative: Function whose derivative gives the integrand
ILATE: Priority rule for integration by parts
Partial fractions: Decomposition of rational functions for integration
Fundamental Theorem: Links definite integral to antiderivative
King's rule: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx

Quick Revision

∫xⁿdx = xⁿ⁺¹/(n+1); ∫e^xdx = e^x; ∫(1/x)dx = ln|x|
By parts: ∫u dv = uv - ∫v du; use ILATE for choosing u
∫e^x[f(x)+f'(x)]dx = e^xf(x) + C
King's rule: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx
Even function: ∫₋ₐᵃ = 2∫₀ᵃ; Odd function: ∫₋ₐᵃ = 0
Partial fractions: degree of numerator must be less than denominator
∫dx/(a²+x²) = (1/a)tan^-1(x/a)