Integrals: The Reverse of Differentiation
Integration is the inverse operation of differentiation. While a derivative finds the rate of change, an integral finds the accumulated quantity. The indefinite integral (antiderivative) finds a family of functions F(x) such that F′(x) = f(x), always including the constant of integration +C. The definite integral computes the signed area under the curve between two bounds. Integrals are essential in physics (work, displacement, flux), engineering (signal processing, control), probability (continuous distributions), and economics (consumer/producer surplus).
Core Integration Rules
Power Rule: ∫x^n dx = x^(n+1)/(n+1) + C (n≠-1)
Constant: ∫k dx = kx + C
Constant Multiple: ∫k·f dx = k·∫f dx
Sum: ∫(f+g) dx = ∫f dx + ∫g dx
Trig Integrals:
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C
Exp/Log:
∫e^x dx = e^x + C
∫1/x dx = ln|x| + C
Definite: ∫[a,b] f(x) dx = F(b) - F(a)
Constant: ∫k dx = kx + C
Constant Multiple: ∫k·f dx = k·∫f dx
Sum: ∫(f+g) dx = ∫f dx + ∫g dx
Trig Integrals:
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C
Exp/Log:
∫e^x dx = e^x + C
∫1/x dx = ln|x| + C
Definite: ∫[a,b] f(x) dx = F(b) - F(a)
Indefinite vs Definite
An indefinite integral gives a function family (+C). A definite integral gives a number (area). The Fundamental Theorem of Calculus connects them: ∫[a,b] f(x)dx = F(b)-F(a) where F is any antiderivative of f.