Integrals
Uttar Pradesh Board · Class 12 · Mathematics
Quick revision notes for Integrals — Uttar Pradesh Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Introduction to Integration
- Integration is the inverse process of differentiation
- If F'(x) = f(x), then ∫f(x)dx = F(x) + C
- Anti-derivative or primitive of a function f is a function F such that F'(x) = f(x)
Standard Integral Formulas
- Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
- Exponential functions: ∫eˣ dx = eˣ + C
- Logarithmic function: ∫(1/x) dx = log|x| + C
Properties of Indefinite Integrals
- Linearity: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
- Scalar multiplication: ∫k·f(x)dx = k∫f(x)dx
- Differentiation and integration are inverse processes
Integration by Substitution
- Used when integrand contains a function and its derivative
- Change of variable: x = g(t), dx = g'(t)dt
- ∫f(x)dx = ∫f(g(t))·g'(t)dt after substitution
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