E Mathematical Formulas - University Physics Volume 1

Quadratic formula

If $a x^{2} + b x + c = 0,$ then $x = \frac{− b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Triangle of base $b$ and height $h$	Area $= \frac{1}{2} b h$
Circle of radius $r$	Circumference $= 2 π r$	Area $= π r^{2}$
Sphere of radius $r$	Surface area $= 4 π r^{2}$	Volume $= \frac{4}{3} π r^{3}$
Cylinder of radius $r$ and height $h$	Area of curved surface $= 2 π r h$	Volume $= π r^{2} h$

Table E1 Geometry

Trigonometry

Trigonometric Identities

Triangles

Series expansions

Binomial theorem: ${(a + b)}^{n} = a^{n} + n a^{n - 1} b + \frac{n (n - 1) a^{n - 2} b^{2}}{2!} + \frac{n (n - 1) (n - 2) a^{n - 3} b^{3}}{3!} + ···$
${(1 \pm x)}^{n} = 1 \pm \frac{n x}{1!} + \frac{n (n - 1) x^{2}}{2!} \pm ··· (x^{2} < 1)$
${(1 \pm x)}^{− n} = 1 \mp \frac{n x}{1!} + \frac{n (n + 1) x^{2}}{2!} \mp ··· (x^{2} < 1)$
$sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - ···$
$cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - ···$
$tan x = x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + ···$
$e^{x} = 1 + x + \frac{x^{2}}{2!} + ···$
$ln (1 + x) = x - \frac{1}{2} x^{2} + \frac{1}{3} x^{3} - ··· (| x | < 1)$

Derivatives

$\frac{d}{d x} [a f (x)] = a \frac{d}{d x} f (x)$
$\frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)$
$\frac{d}{d x} [f (x) g (x)] = f (x) \frac{d}{d x} g (x) + g (x) \frac{d}{d x} f (x)$
$\frac{d}{d x} f (u) = [\frac{d}{d u} f (u)] \frac{d u}{d x}$
$\frac{d}{d x} x^{m} = m x^{m - 1}$
$\frac{d}{d x} sin x = cos x$
$\frac{d}{d x} cos x = − sin x$
$\frac{d}{d x} tan x = {sec}^{2} x$
$\frac{d}{d x} cot x = − {csc}^{2} x$
$\frac{d}{d x} sec x = tan x sec x$
$\frac{d}{d x} csc x = − cot x csc x$
$\frac{d}{d x} e^{x} = e^{x}$
$\frac{d}{d x} ln x = \frac{1}{x}$
$\frac{d}{d x} {sin}^{−1} x = \frac{1}{\sqrt{1 - x^{2}}}$
$\frac{d}{d x} {cos}^{−1} x = - \frac{1}{\sqrt{1 - x^{2}}}$
$\frac{d}{d x} {tan}^{−1} x = \frac{1}{1 + x^{2}}$

Integrals

$\int a f (x) d x = a \int f (x) d x$
$\int [f (x) + g (x)] d x = \int f (x) d x + \int g (x) d x$
$\begin{matrix} \int x^{m} d x & = \frac{x^{m + 1}}{m + 1} (m \neq − 1) \\ = ln x (m = −1) \end{matrix}$
$\int sin x d x = − cos x$
$\int cos x d x = sin x$
$\int tan x d x = ln | sec x |$
$\int {sin}^{2} a x d x = \frac{x}{2} - \frac{sin 2 a x}{4 a}$
$\int {cos}^{2} a x d x = \frac{x}{2} + \frac{sin 2 a x}{4 a}$
$\int sin a x cos a x d x = - \frac{cos 2 a x}{4 a}$
$\int e^{a x} d x = \frac{1}{a} e^{a x}$
$\int x e^{a x} d x = \frac{e^{a x}}{a^{2}} (a x - 1)$
$\int ln a x d x = x ln a x - x$
$\int \frac{d x}{a^{2} + x^{2}} = \frac{1}{a} {tan}^{−1} \frac{x}{a}$
$\int \frac{d x}{a^{2} - x^{2}} = \frac{1}{2 a} ln | \frac{x + a}{x - a} |$
$\int \frac{d x}{\sqrt{a^{2} + x^{2}}} = {sinh}^{−1} \frac{x}{a}$
$\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = {sin}^{−1} \frac{x}{a}$
$\int \sqrt{a^{2} + x^{2}} d x = \frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} {sinh}^{−1} \frac{x}{a}$
$\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} {sin}^{−1} \frac{x}{a}$

E | Mathematical Formulas