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Formula: \frac{e^{i\theta}-e^{-i\theta}}2
Page: 222
Section: 2
Chapter: 4
Notation: \sin
Taylor series: x\mapsto\sum\frac{(-1)^k x^{2k+1} }{(2k+1)!}
Radius: \(+\infty\)
Monotonic: No
Extra: Ahlfor, complex analysis: ch 2. Sec. 3.2. Page 44
Visual complex analysis: ch 2V, Section 1, page 84
Zorich Ch 5. 5.4 P 271
Real analysis, Pugh: ch4 sec 2. Page 222
Parity: odd

Formula: \pm i\log\left(w+\sqrt{w^ 2-1}\right)
Page: 047
Section: 3.4
Chapter: 2
Notation: \arccos

Formula: \frac{e^{iz}-e^{-iz} }{2}
Page: 271,272
Section: 5.3
Chapter: 5
Notation: \sin
Radius: \(+\infty\)
Parity: An odd function

Formula: \frac{1}{(x+a)^2}
Integral: -\frac{1}{x+a}
Derivative: \frac{-2}{(a+x)^{-3}}
Taylor series: \sum_i frac{(-ax)^i}{b^i+1}
Radius: \(\left|\frac ba\right|\)
Nth derivative: \frac{(-1-m)_m}{(a+x)^{-2-m} }\text{ for }x\ne -2,x\ne 1
Assuming: \(x\ne -2,x\ne 1\)

Formula: x^n
Integral: \frac{x^{n+1}}{n+1}, \text{for $n\ne 1$}
Derivative: nx^{n-1}
Radius: \(+\infty\)
Nth derivative: \left\{\begin{array}{ll}(n)_mx^{n-m}& \text{if } n\ne \mathbb Z\lor m\le n\\0&\text{otherwise}\end{array}\right.

Formula: \frac{1}{x}
Integral: \ln|x|
Derivative: -(\frac{-1}{x})^{2}
Taylor series: none
Nth derivative: -(\frac{-1}{x})^{m+1}m!

Formula: \frac{1}{ax+b}
Integral: \frac{1}{a}\ln\left|ax + b\right|
Derivative: \frac{a}{(ax+b)^{2}}
Taylor series: \sum_{i=0}^\infty(-1)^i\frac{a^ix^i}{b^{i+1} }
Radius: \(\left|\frac ba\right|\)
Nth derivative: \frac{m!a^{m}}{(ax+b)^{m+1}}

Formula: \(\frac{1}{(x+a)^2}\)
Integral: -\frac{1}{x+a}

Formula: (x+a)^n
Integral: \frac{(x+a)^{n+1}}{n+1}, n\ne -1
Derivative: n(x+a)^{n-1}
Taylor series: \sum_{i=0}^na^{n-i}{n\choose i}x^i
Radius: \(\infty\)
Nth derivative: (n)_m(x+a)^{n-m}

Formula: x(x+a)^n
Integral: \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)}
Derivative: (a+x)^{n-1}(a+nx+x)
Taylor series: \sum_{i=0}^na^{n-i}{n\choose i}x^{i+1}
Radius: \(\infty\)
Nth derivative: (n)_{m-1}(a+x)^{n-m}(ma+nx+x)

Formula: \(\frac{1}{1+x^2}\)
Integral: \tan^{-1}x

Formula: \(\frac{1}{a^2+x^2}\)
Integral: \frac{1}{a}\tan^{-1}\frac{x}{a}

Formula: \(\frac{x}{a^2+x^2}\)
Integral: \frac{1}{2}\ln|a^2+x^2|

Formula: \(\frac{x^2}{a^2+x^2}\)
Integral: x-a\tan^{-1}\frac{x}{a}

Formula: \(\frac{x^3}{a^2+x^2}\)
Integral: \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2|

Formula: \(\frac{1}{ax^2+bx+c}\)
Integral: \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}

Formula: \(\frac{1}{(x+a)(x+b)}\)
Integral: \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b

Formula: \(\frac{x}{(x+a)^2}\)
Integral: \frac{a}{a+x}+\ln |a+x|

Formula: \(\frac{x}{ax^2+bx+c}\)
Integral: \frac{1}{2a}\ln|ax^2+bx+c| -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}

Formula: \sqrt{x-a}
Integral: \frac{2}{3}(x-a)^{3/2}
Derivative: \frac{1}{2\sqrt{x-a}}

Formula: \(\frac{1}{\sqrt{x\pm a}}\)
Integral: 2\sqrt{x\pm a}

Formula: \frac{1}{\sqrt{a-x}}
Integral: -2\sqrt{a-x}

Formula: \(x\sqrt{x-a}\)
Integral: \left\{\begin{array}{l}\frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or} \\ \frac{2}{3} x(x-a)^{3/2} - \frac{4}{15} (x-a)^{5/2}, \text{ or}\\ \frac{2}{15}(2a+3x)(x-a)^{3/2}\end{array}\right.

Formula: \(\sqrt{ax+b}\)
Integral: \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}

Formula: \((ax+b)^{3/2}\)
Integral: \frac{2}{5a}(ax+b)^{5/2}

Formula: \(\frac{x}{\sqrt{x\pm a} } \)
Integral: \frac{2}{3}(x\mp 2a)\sqrt{x\pm a}

Formula: \(\sqrt{\frac{x}{a-x}}\)
Integral: -\sqrt{x(a-x)} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}

Formula: \(\sqrt{\frac{x}{a+x}}\)
Integral: \sqrt{x(a+x)} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right]

Formula: \(x \sqrt{ax + b}\)
Integral: \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2)\sqrt{ax+b}

Formula: \(\sqrt{x(ax+b)}\)
Integral: \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)}-b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ]

Formula: \(\sqrt{x^3(ax+b)} \)
Integral: \left [ \frac{b}{12a}-\frac{b^2}{8a^2x}+\frac{x}{3}\right] \sqrt{x^3(ax+b)} + \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right |

Formula: \(\sqrt{x^2 \pm a^2} \)
Integral: \frac{1}{2}x\sqrt{x^2\pm a^2} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right |

Formula: \(\sqrt{a^2 - x^2} \)
Integral: \frac{1}{2} x \sqrt{a^2-x^2} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}

Formula: \(x \sqrt{x^2 \pm a^2}\)
Integral: \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2}

Formula: \(\frac{1}{\sqrt{x^2 \pm a^2}} \)
Integral: \ln \left | x + \sqrt{x^2 \pm a^2} \right |

Formula: \(\frac{1}{\sqrt{a^2 - x^2}} \)
Integral: \sin^{-1}\frac{x}{a}

Formula: \(\frac{x}{\sqrt{x^2\pm a^2}} \)
Integral: \sqrt{x^2 \pm a^2}

Formula: \(\frac{x}{\sqrt{a^2-x^2}} \)
Integral: -\sqrt{a^2-x^2}

Formula: \(\frac{x^2}{\sqrt{x^2 \pm a^2}} \)
Integral: \frac{1}{2}x\sqrt{x^2 \pm a^2}\mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right |

Formula: \(\sqrt{a x^2 + b x + c} \)
Integral: \frac{b+2ax}{4a}\sqrt{ax^2+bx+c}+\frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right |

Formula: \(x \sqrt{a x^2 + bx + c} \)
Integral: \[\frac{1}{48a^{5/2}}\begin{split}& \left ( 2 \sqrt{a} \sqrt{ax^2+bx+c} \right . \left( - 3b^2 + 2 abx + 8 a(c+ax^2) \right) \\ & \left. + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \end{split}\]

Formula: \(\frac{1}{\sqrt{ax^2+bx+c}}\)
Integral: \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |

Formula: \(\frac{x}{\sqrt{ax^2+bx+c}}\)
Integral: \frac{1}{a}\sqrt{ax^2+bx + c} -\frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |

Formula: \(\frac{1}{(a^2+x^2)^{3/2}}\)
Integral: \frac{x}{a^2\sqrt{a^2+x^2}}

Formula: \(\ln ax\)
Integral: x \ln ax - x

Formula: \(x \ln x \)
Integral: \frac{1}{2} x^2 \ln x-\frac{x^2}{4}

Formula: \(x^2 \ln x \)
Integral: \frac{1}{3} x^3 \ln x-\frac{x^3}{9}

Formula: \(x^n \ln x \)
Integral: x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1

Formula: \(\frac{\ln ax}{x} \)
Integral: \frac{1}{2}\left ( \ln ax \right)^2

Formula: \(\frac{\ln x}{x^2} \)
Integral: -\frac{1}{x}-\frac{\ln x}{x}

Formula: \(\ln (1-x) \)
Integral: \left ( x + -1 \right) \ln (1-x) - x
Taylor series: -\sum_{n=1}^\infty\frac{x^n}n
Radius: \(1\)
Nth derivative: \frac{(-1)^{m+1}(m-1)!(-1)^{m}}{(1-x)^{m}}

Formula: \(\ln ( x^2 + a^2 ) \)
Integral: x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} - 2x

Formula: \(\ln ( x^2 - a^2 ) \)
Integral: x \ln (x^2 - a^2 ) +a\ln \frac{x+a}{x-a} - 2x

Formula: \(\ln \left ( ax^2 + bx + c\right) \)
Integral: \frac{1}{a}\sqrt{4ac-b^2}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}-2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right)

Formula: \(x \ln (ax + b) \)
Integral: \frac{bx}{2a}-\frac{1}{4}x^2 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b)

Formula: \(x\ln\left( a^2 - b^2 x^2\right)\)
Integral: -\frac{1}{2}x^2+ \frac{1}{2}\left( x^2 - \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right)

Formula: \((\ln x)^2 \)
Integral: 2x - 2x \ln x + x (\ln x)^2

Formula: \((\ln x)^3 \)
Integral: -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x

Formula: \(x (\ln x)^2 \)
Integral: \frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x

Formula: \(x^2 (\ln x)^2 \)
Integral: \frac{2 x^3}{27}+\frac{1}{3} x^3 (\ln x)^2-\frac{2}{9} x^3 \ln x

Formula: \lim_{n\to\infty}\left(1+\frac zn\right)^n
Integral: e^{x}
Notation: e^x
Name: Exponential
Taylor series: \sum_{n=0}^\infty\frac{x^n}{n!}
Radius: \(\infty\)
Extra: Visual complex analysis: 2IV. 3. 82
Complex analysis, Ahlfor, 2. 3.1. 43
Real analysis, Pugh, 4. 2. 222
Parametrized complexity 1I, 5. 11.
Definition: the only function \(f\) such that \(f(0)=1\) and \(f'=f\)
Inverse: logarithm

Formula: \(\sqrt{x} e^{ax} \)
Integral: \frac{1}{a}\sqrt{x}e^{ax} +\frac{i\sqrt{\pi}}{2a^{3/2}}\text{erf}\left(i\sqrt{ax}\right),\text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}d t

Formula: \(x e^x \)
Integral: (x-1) e^x

Formula: \(x e^{ax} \)
Integral: \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax}

Formula: \(x^2 e^{x} \)
Integral: \left(x^2 - 2x + 2\right) e^{x}

Formula: \(x^2 e^{ax} \)
Integral: \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax}

Formula: \(x^3 e^{x} \)
Integral: \left(x^3-3x^2 + 6x - 6\right) e^{x}

Formula: \(x^n e^{ax} \)
Integral: \dfrac{x^n e^{ax}}{a} - \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt} d x

Formula: \(e^{ax^2} \)
Integral: -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right)

Formula: \(e^{-ax^2} \)
Integral: \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right)

Formula: \(x e^{-ax^2} \)
Integral: -\dfrac{1}{2a}e^{-ax^2}

Formula: \(x^2 e^{-ax^2} \)
Integral: \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2}

Integral: -\cos x
Notation: \sin
Derivative: \cos x
Taylor series: \sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}
Radius: \(\infty\)
Nth derivative: \sin(x+\frac{m\pi}2)=(-1)^{\frac m2}f(x) \text{ with $f$ being $\sin$ if $m$ is even, and $\cos$ otherwise}

Formula: \(\sin^2 ax \)
Integral: \frac{x}{2} - \frac{\sin 2ax} {4a}

Formula: \(\sin^3 ax \)
Integral: -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a}

Formula: \(\sin^n ax \)
Integral: -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[\frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax\right]

Integral: \sin x
Notation: \cos
Derivative: \cos x
Taylor series: \sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}
Nth derivative: \cos(x+\frac{m\pi}2)=(-1)^{\frac {m+1}2}f(x) \text{ with $f$ being $\cos$ if $m$ is even, and $\sin$ otherwise}

Formula: \(\cos^2 ax \)
Integral: \frac{x}{2}+\frac{ \sin 2ax}{4a}

Formula: \(\cos^3 ax\)
Integral: \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a}

Formula: \(\cos^p ax\)
Integral: -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[\frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax\right]

Formula: \(\cos x \sin x \)
Integral: \frac{1}{2}\sin^2 x + c_1 = -\frac{1}{2} \cos^2x + c_2 = -\frac{1}{4} \cos 2x + c_3

Formula: \(\cos ax \sin bx \)
Integral: \frac{\cos[(a-b) x]}{2(a-b)} - \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b

Formula: \(\sin^2 ax \cos bx \)
Integral: -\frac{\sin[(2a-b)x]}{4(2a-b)} + \frac{\sin bx}{2b} - \frac{\sin[(2a+b)x]}{4(2a+b)}

Formula: \(\sin^2 x \cos x \)
Integral: \frac{1}{3} \sin^3 x

Formula: \(\cos^2 ax \sin bx \)
Integral: \frac{\cos[(2a-b)x]}{4(2a-b)} - \frac{\cos bx}{2b} - \frac{\cos[(2a+b)x]}{4(2a+b)}

Formula: \(\cos^2 ax \sin ax \)
Integral: -\frac{1}{3a}\cos^3{ax}

Formula: \(\sin^2 ax \cos^2 bx\)
Integral: \frac{x}{4}-\frac{\sin 2ax}{8a}-\frac{\sin[2(a-b)x]}{16(a-b)}+\frac{\sin 2bx}{8b}-\frac{\sin[2(a+b)x]}{16(a+b)}

Formula: \(\sin^2 ax \cos^2 ax \)
Integral: \frac{x}{8}-\frac{\sin 4ax}{32a}

Formula: \(\tan ax \)
Integral: -\frac{1}{a} \ln \cos ax

Formula: \(\tan^2 ax \)
Integral: -x + \frac{1}{a} \tan ax

Formula: \(\tan^n ax \)
Integral: \frac{\tan^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\tan^2 ax \right)

Formula: \(\tan^3 ax\)
Integral: \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax

Integral: \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right)
Notation: \sec

Formula: \(\sec^2 ax \)
Integral: \frac{1}{a} \tan ax

Formula: \(\sec^3 x \)
Integral: \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x |

Formula: \(\sec x \tan x \)
Integral: \sec x

Formula: \(\sec^2 x \tan x \)
Integral: \frac{1}{2} \sec^2 x

Formula: \(\sec^n x \tan x \)
Integral: \frac{1}{n} \sec^n x , n\ne 0

Integral: \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x - \cot x| + C
Notation: \csc

Formula: \(\csc^2 ax \)
Integral: -\frac{1}{a} \cot ax

Formula: \(\csc^3 x \)
Integral: -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x - \cot x |

Formula: \(\csc^nx \cot x \)
Integral: -\frac{1}{n}\csc^n x, n\ne 0

Formula: \(\sec x \csc x \)
Integral: \ln | \tan x |

Formula: \(x \cos x \)
Integral: \cos x + x \sin x

Formula: \(x \cos ax \)
Integral: \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax

Formula: \(x^2 \cos x \)
Integral: 2 x \cos x + \left ( x^2 - 2 \right ) \sin x

Formula: \(x^2 \cos ax \)
Integral: \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 - 2 }{a^3} \sin ax

Formula: \(x^n \cos x\)
Integral: -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) + (-1)^n \Gamma(n+1, ix)\right]

Formula: \(x^n \cos ax \)
Integral: \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) -\Gamma(n+1, ixa)\right]

Formula: \(x \sin x \)
Integral: -x \cos x + \sin x

Formula: \(x \sin ax \)
Integral: -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2}

Formula: \(x^2 \sin x \)
Integral: \left(2-x^2\right) \cos x + 2 x \sin x

Formula: \(x^2 \sin ax \)
Integral: \frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2}

Formula: \(x^n \sin x \)
Integral: -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) - (-1)^n\Gamma(n+1, -ix)\right]

Formula: \(x \cos^2 x \)
Integral: \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x

Formula: \(x \sin^2 x \)
Integral: \frac{x^2}{4}-\frac{1}{8}\cos 2x - \frac{1}{4} x \sin 2x

Formula: \(x \tan^2 x \)
Integral: -\frac{x^2}{2} + \ln \cos x + x \tan x

Formula: \(x \sec^2 x \)
Integral: \ln \cos x + x \tan x

Formula: \(e^x \sin x \)
Integral: \frac{1}{2}e^x (\sin x - \cos x)

Formula: \(e^{bx} \sin ax \)
Integral: \frac{1}{a^2+b^2}e^{bx} (b\sin ax - a\cos ax)

Formula: \(e^x \cos x \)
Integral: \frac{1}{2}e^x (\sin x + \cos x)

Formula: \(e^{bx} \cos ax \)
Integral: \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax )

Formula: \(x e^x \sin x \)
Integral: \frac{1}{2}e^x (\cos x - x \cos x + x \sin x)

Formula: \(x e^x \cos x \)
Integral: \frac{1}{2}e^x (x \cos x - \sin x + x \sin x)

Formula: \(\cosh ax \)
Integral: \frac{1}{a} \sinh ax

Formula: e^{ax}  \cosh bx
Integral: \begin{cases}\displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx - b \sinh bx ] & a\ne b \\\displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b\end{cases}

Formula: \(\sinh ax \)
Integral: \frac{1}{a} \cosh ax

Formula: \(e^{ax} \sinh bx \)
Integral: \begin{cases}\displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\\displaystyle{\frac{e^{2ax}}{4a} - \frac{x}{2}} & a = b\end{cases}

Formula: \(\tanh ax\)
Integral: \frac{1}{a} \ln \cosh ax

Formula: \(e^{ax} \tanh bx \)
Integral: \begin{cases}\displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\\displaystyle{\hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right]} & a\ne b \\\displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b\end{cases}

Formula: \(\cos ax \cosh bx \)
Integral: \frac{1}{a^2 + b^2} \left[a \sin ax \cosh bx + b \cos ax \sinh bx\right]

Formula: \(\cos ax \sinh bx \)
Integral: \frac{1}{a^2 + b^2} \left[b \cos ax \cosh bx + a \sin ax \sinh bx\right]

Formula: \(\sin ax \cosh bx \)
Integral: \frac{1}{a^2 + b^2} \left[-a \cos ax \cosh bx + b \sin ax \sinh bx\right]

Formula: \(\sin ax \sinh bx \)
Integral: \frac{1}{a^2 + b^2} \left[b \cosh bx \sin ax - a \cos ax \sinh bx\right]

Formula: \(\sinh ax \cosh ax\)
Integral: \frac{1}{4a}\left[ -2ax + \sinh 2ax \right]

Formula: \(\sinh ax \cosh bx \)
Integral: \frac{1}{b^2-a^2}\left[ b \cosh bx \sinh ax 3- a \cosh ax \sinh bx \right]

Formula: \(e^{ax} \)
Integral: \frac{1}{a}e^{ax}
Taylor series: \sum_{n=0}^\infty\frac {x^n}{n!}
Nth derivative: a^me^{ax}

Formula: \(\sin ax \)
Integral: -\frac{1}{a} \cos ax

Formula: \frac1{1-x}
Integral: -1 \ln |1-x|
Derivative: \frac{1}{1-x}
Taylor series: \sum_{n=0}^\infty x^n
Radius: \(1\)
Nth derivative: (-1)^{m-1}\frac{m!}{(1-x)^m}

Formula: \ln(ax+b)
Integral: \left(x+\frac{b}{a}\right)\ln(ax+b)-x
Derivative: \frac{a}{(ax+b)}
Taylor series: \ln b+\sum _{n=1}^\infty(-1)^{n+1}\frac{(ax)^n}{nb}
Radius: \(\left|\frac ba\right|\)
Nth derivative: \frac{(-1)^{m+1}(m-1)!a^{m}}{(ax+b)^{m}}
Assuming: \(a\ne 0\)

Formula: \ln(1+x)
Integral: \left(x+1\right)\ln(1+x)-x
Derivative: \frac{1}{x+1}
Taylor series: \sum_{n=1}^\infty(-1)^{n+1}\frac{x^n}n
Radius: \(1\)
Nth derivative: (-1)^{m+1}\frac{(m-1)!}{(x+1)^{m}}

Formula: a^x
Integral: \frac{a^x}{\ln a}
Name: Exponential in base \(a\)

Page: 193
Section: 2
Chapter: 5
Notation: \cot
Index: 3
Kind: Ex
Derivative: -\frac1{\sin^2x}

Notation: \arcsin
Derivative: \frac{1}{\sqrt{1-x^2}}

Notation: \tanh
Derivative: 1/cosh²

Notation: \coth
Derivative: -\frac{1}{\sinh^2}

Formula: \(\frac{1}{\sqrt{x^2\pm1}}\)
Integral: \ln\left|x+\sqrt{x^2\pm1}\right|

Formula: \frac{1}{1-x^2}
Integral: \frac12\ln\left|\frac{1+x}{1-x}\right|

Formula: \int_1^x\frac{e^t}{t} d t
Page: 591
Section: 1.1
Chapter: 19
Notation: Ei(x)
Name: The exponential integral function
Index: 7
Kind: Ex

Formula: \int_{1}^x\frac 1t d t
Integral: x\ln x-x
Page: 186
Section: 2
Chapter: 3
Notation: \ln
Name: The natural logarithm function
Derivative: \frac 1t
Inverse: \(e^x\)

Formula: \int_0^x\frac{\sin t}{t} d t
Integral: x\text{Si}(x)+\cos x+C
Page: 314, https://calculus.subwiki.org/wiki/Sine_integral https://en.wikipedia.org/wiki/Trigonometric_integral
Section: 7.2c
Chapter: 5
Notation: \text{Si}
Name: The sine integral
Index: 32
Kind: Ex
Taylor series: \sum_{k\in\mathbb N}\frac{(-1)^kx^{2k+1}}{(2k+1)(2k+1)!}
Context: Analysis

Formula: \int_0^x\frac{1}{\ln t}d t
Integral: z\mathtt{li}(z) - Ei(2\ln z)
Page: 314, https://en.wikipedia.org/wiki/Logarithmic_integral_function
Section: 7.2c
Chapter: 5
Variables: with \(EI\) the exponential integral
Notation: Li(x)
Name: Logarithmic integral function
Derivative: \frac{1}{\ln t}
Domain: \([0,1)\)
Context: Analysis

Page: 306
Section: 7.1
Chapter: 5
Notation: \arctan
Derivative: \frac{1}{1+x^2}
Extra: equals to \(\mathtt{arccot}\frac1x\)
Context: Analysis

Formula: \(E(e^{tX})\)
Page: 75
Chapter: 5
Variables: \(X\) a random variable
Notation: \(M_X(t)\)
Name: The moment generating function of \(X\)
Taylor series: \sum_{n=0}^\infty\frac{t^n \mu'_n}{n!}
Domain: A real function
Abbreviation: m.g.f. of \(X\)