Algebra

Geometry and Trigonometry

Binomial expansion:	${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2} x^{2} + \dots$
Binomial approximation:	${(1 + x)}^{n} \approx 1 + n x if x ≪ 1$
Trigonometric expansions:	$sin α = α - \frac{α^{3}}{3!} + \frac{α^{5}}{5!} - \frac{α^{7}}{7!} + \dots$ for $α$ in rad
	$cos α = 1 - \frac{α^{2}}{2!} + \frac{α^{4}}{4!} - \frac{α^{6}}{6!} + \dots$ for $α$ in rad
Small-angle approximation:	If $α ≪ 1$ rad, then $sin α \approx tan α \approx α$ and $cos α \approx 1$ .
	The small-angle approximation is excellent for $α < 5 °$ ( $\approx 0.1$ rad) and generally acceptable up to $α \approx 10 °$ .

The letters a and n represent constants in the following derivatives and integrals.

$\int_{}^{} x d x = \frac{1}{2} x^{2}$	$\int_{}^{} \frac{d x}{{(x^{2} \pm a^{2})}^{3 / 2}} = \frac{\pm x}{a^{2} \sqrt{x^{2} \pm a^{2}}}$

$\int_{}^{} x^{2} d x = \frac{1}{3} x^{3}$	$\int_{}^{} \frac{x d x}{{(x^{2} \pm a^{2})}^{3 / 2}} = - \frac{1}{\sqrt{x^{2} \pm a^{2}}}$

$\int_{}^{} \frac{1}{x^{2}} d x = - \frac{1}{x}$	$\int_{}^{} e^{a x} d x = \frac{1}{a} e^{a x}$

$\int_{}^{} x^{n} d x = \frac{x^{n + 1}}{n + 1} n \neq - 1$	$\int_{}^{} x e^{- x} d x = - (x + 1) e^{- x}$

$\int_{}^{} \frac{d x}{x} = ln x$	$\int_{}^{} x^{2} e^{- x} d x = - (x^{2} + 2 x + 2) e^{- x}$

$\int_{}^{} \frac{d x}{a + x} = ln (a + x)$	$\int_{}^{} sin (a x) d x = - \frac{1}{a} cos (a x)$

$\int_{}^{} \frac{x d x}{a + x} = x - a ln (a + x)$	$\int_{}^{} cos (a x) d x = \frac{1}{a} sin (a x)$

$\int_{}^{} \frac{d x}{\sqrt{x^{2} \pm a^{2}}} = ln (x + \sqrt{x^{2} \pm a^{2}})$	$\int_{}^{} {sin}^{2} (a x) d x = \frac{x}{2} - \frac{sin (2 a x)}{4 a}$

$\int_{}^{} \frac{x d x}{\sqrt{x^{2} \pm a^{2}}} = \sqrt{x^{2} \pm a^{2}}$	$\int_{}^{} {cos}^{2} (a x) d x = \frac{x}{2} + \frac{sin (2 a x)}{4 a}$

$\int_{}^{} \frac{d x}{x^{2} + a^{2}} = \frac{1}{a} {tan}^{- 1} (\frac{x}{a})$	$\int_{0}^{\infty} x^{n} e^{- a x} d x = \frac{n!}{a^{n + 1}}$

$\int_{}^{} \frac{d x}{{(x^{2} + a^{2})}^{2}} = \frac{1}{2 a^{3}} {tan}^{- 1} (\frac{x}{a}) + \frac{x}{2 a^{2} (x^{2} + a^{2})}$	$\int_{0}^{\infty} e^{- a x^{2}} d x = \frac{1}{2} \sqrt{\frac{π}{a}}$

Using exponents:	$a^{- x} = \frac{1}{a^{x}}$	$a^{x} a^{y} = a^{(x + y)}$	$\frac{a^{x}}{a^{y}} = a^{(x - y)}$ ${(a^{x})}^{y} = a^{x y}$

	$a^{0} = 1$	$a^{1} = a$	$a^{1 / n} = \sqrt[n]{a}$

Fractions:	$(\frac{a}{b}) (\frac{c}{d}) = \frac{a c}{b d}$	$\frac{a / b}{c / d} = \frac{a d}{b c}$	$\frac{1}{1 / a} = a$

Logarithms:	If $a = e^{x}$ , then $ln (a) = x$		$ln (e^{x}) = x$	$e^{ln (x)} = x$

	$ln (a b) = ln (a) + ln (b)$		$ln (\frac{a}{b}) = ln (a) - ln (b)$	$ln (a^{n}) = n ln (a)$
	The expression $ln(a + b)$ cannot be simplified.
Quadratic equation:	The quadratic equation $a x^{2} + b x + c = 0$ has the two solutions $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .
Linear equations:	The graph of the equation $y = a x + b$ is a straight line. a is the slope of the graph. b is the y-intercept.
Proportionality:	To say that y is proportional to x, written $y \propto x$ , means that $y = a x$ , where a is a constant. Proportionality is a special case of linearity. A graph of a proportional relationship is a straight line that passes through the origin. If $y \propto x$ , then
	$\frac{y_{1}}{y_{2}} = \frac{x_{1}}{x_{2}}$
Distance:	The distance between points with coordinates $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Area and volume:	Rectangle $A = a b$	Rectangular box $V = a b c$
	Triangle $A = \frac{1}{2} a b$	Right circular cylinder $V = π r^{2} l$
	Circle $C = 2 π r$ $A = π r^{2}$	Sphere $A = 4 π r^{2}$ $V = \frac{4}{3} π r^{3}$

Arc length and angle:	The angle $θ$ in radians is defined as $θ = s / r$ . The arc length that spans angle $θ$ is $s = r θ$ . $2 π rad = 360 °$
Right triangle:	Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ or $a^{2} + b^{2} = c^{2}$ $\begin{array}{l} sin θ = \frac{b}{c} = \frac{far side}{hypotenuse} & θ = {sin}^{- 1} (\frac{b}{c}) \\ cos θ = \frac{a}{c} = \frac{adjacent side}{hypotenuse} & θ = {cos}^{- 1} (\frac{a}{c}) \\ tan θ = \frac{b}{a} = \frac{far side}{adjacent side} & θ = {tan}^{- 1} (\frac{b}{a}) \end{array}$
General triangle:	$α + β + γ = 180 ° = π rad$ Law of cosines $c^{2} = a^{2} + b^{2} - 2 a b cos γ$
Identities:	$tan α = \frac{sin α}{cos α}$	${sin}^{2} α + {cos}^{2} α = 1$
	$sin (- α) = - s i n α$	$cos (- α) = c o s α$
	$sin (α \pm β) = s i n α c o s β \pm c o s α s i n β$	$cos (α \pm β) = c o s α c o s β \mp s i n α s i n β$
	$sin (2 α) = 2 sin α c o s α$	$cos (2 α) = {cos}^{2} α - {sin}^{2} α$
	$sin (α \pm π / 2) = \pm cos α$	$cos (α \pm π / 2) = \mp sin α$
	$sin (α \pm π) = - s i n α$	$cos (α \pm π) = - c o s α$
Trigonometric functions:	$\sin θ and cos θ$ are the functions of the angle $θ$ with period $2 π rad$ .