2.3 Integrals

Definition

Indefinite Integral. A function F(x) is called an antiderivative of a continuous function ƒ(x) if F(x)′=ƒ(x). Indefinite integral of ƒ(x) represents a family of all functions F(x)+C, where C is a constant.

Definite Integral. Real number I is called a definite integral of a continuous ƒ(x) on the interval [a,b] if I=(lim_n→∞)((∑_i=0^N)(ƒ((x_i))*Δ(x))).

Notations.

Fundamental Theorem of Calculus

Part I

If ƒ is continous on [a,b] and F(x)=(∫_a^x)(ƒ(t)*d(t)), then (F^′)(x)=ƒ(x)

Part II

Ifƒ is continous on [a,b] and F is an antiderivative of ƒ, then (∫_a^b)(ƒ(x)*d(x))=F(b)-F(a)

Rules

Commutation: ∫(ƒ(x)+g(x))*d(x)=∫(ƒ(x)*d(x))+∫(g(x)*d(x))

Product: ∫(ƒ(x)*g(x)*d(x))=∫(ƒ(x))*d(x)⋅g(x)+∫(g(x)*d(x))⋅ƒ(x)

(∫_a^a)(ƒ(x)*d(x))=0

(∫_a^b)(ƒ(x)*d(x))=-(∫_b^a)(ƒ(x)*d(x))

Known Indefinite Integrals

Integration Techniques

Substitution

Let u=g(x), then d(u)=(g^′)(x)*d(x). Replace∫(ƒ(g(x))*(g^′)(x)*d(x))=∫(ƒ(u)*d(u)).

Integration by Parts

(∫_^)(u*d(v))=u*v-(∫_^)(v*d(u))

Partial Fractions

Decompose rational P(x)/Q(x) into simpler fractions; integrate term‑by‑term.

Trig Substitution

Use x=sin(θ), tan(θ),sec(θ) to simplify √(,a2±x2).

Improper Integrals

Check limits: (∫_a^∞)(ƒ(x)*d(x))=(lim_b→∞)((∫_a^b)(ƒ(x)*d(x)))

Applications of Integrals

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