2.3 Integrals

Definition

Indefinite Integral. A function $F\left(x\right)$ is called an antiderivative of a continuous function $ƒ\left(x\right)$ if ${F\left(x\right)}^{\prime }=ƒ\left(x\right)$. Indefinite integral of $ƒ\left(x\right)$ represents a family of all functions $F\left(x\right)+C$, where $C$ is a constant.

Definite Integral. Real number $I$ is called a definite integral of a continuous $ƒ\left(x\right)$ on the interval $[a,b]$ if $I={\lim }_{n\to \infty }{\sum }_{\text{i}=0}^{\text{N}}ƒ\left(x_{\text{i}}\right)\Delta x$.

Notations.

Fundamental Theorem of Calculus

Part I

If $ƒ$ is continous on $[a,b]$ and $F\left(x\right)={\int }_a^x(ƒ\left(t\right)dt)$, then $F^{\prime }\left(x\right)=ƒ\left(x\right)$

Part II

If$ƒ$ is continous on $[a,b]$ and $F$ is an antiderivative of $ƒ$, then ${\int }_a^b(ƒ\left(x\right)dx)=F\left(b\right)-F\left(a\right)$

Rules

Commutation: $\int (ƒ\left(x\right)+g\left(x\right))dx=\int ƒ\left(x\right)dx+\int g\left(x\right)dx$

Product: $\int ƒ\left(x\right)g\left(x\right)dx=\int (ƒ\left(x\right))dx\cdot g\left(x\right)+\int g\left(x\right)dx\cdot ƒ\left(x\right)$

${\int }_a^aƒ\left(x\right)dx=0$

${\int }_a^bƒ\left(x\right)dx=-{\int }_b^aƒ\left(x\right)dx$

Known Indefinite Integrals

Integration Techniques

Substitution

Let $u=g\left(x\right)$, then $du=g^{\prime }\left(x\right)dx$. Replace$\int (ƒ(g\left(x\right))g^{\prime }\left(x\right)dx)=\int (ƒ\left(u\right)du)$.

Integration by Parts

${\int }_{}^{}udv=uv-{\int }_{}^{}vdu$

Partial Fractions

Decompose rational $\frac{P\left(x\right)}{Q\left(x\right)}$ into simpler fractions; integrate term‑by‑term.

Trig Substitution

Use $x=\sin \theta$, $\tan \theta ,\sec \theta$ to simplify $\sqrt{a^2\pm x^2}$.

Improper Integrals

Check limits: ${\int }_a^{\infty }ƒ\left(x\right)dx={\lim }_{b\to \infty }{\int }_{\text{a}}^bƒ\left(x\right)dx$

Applications of Integrals

More Cheat Sheets ->