MathematicsDerivative, Integral & Laplace Tables

Derivative Table

  • xx is the independent variable
  • uu and vv are dependent on xx
  • ww is dependent on uu
  • aa and nn are constants
  • log⁡x\log{x} is the common logarithm to base 10, log⁡10x\log_{10}{x}
  • ln⁡x\ln{x} is logarithm to the base ee, log⁡ex\log_e{x}
Fundamental Derivatives
dadx=0\frac{da}{dx} = 0
d(ax)dx=a\frac{d\left(ax\right)}{dx} = a
dxndx=nxn−1\frac{dx^n}{dx} = nx^{n-1}
d(u+v)dx=dudx+dvdx\frac{d\left(u + v \right)}{dx} = \frac{du}{dx} + \frac{dv}{dx}
d(uv)dx=udvdx+vdudx\frac{d\left(uv \right)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}
d(uv)dx=1v2(vdudx−udvdx)\frac{d\left(\frac{u}{v} \right)}{dx} = \frac{1}{v^2} \left( v\frac{du}{dx} - u\frac{dv}{dx} \right)
dwdx=dwdududx\frac{dw}{dx} = \frac{dw}{du}\frac{du}{dx}
dundx=nun−1dudx\frac{du^n}{dx} = nu^{n-1}\frac{du}{dx}
Expressions Containing Exponential and Logarithmic Functions
dln⁡xdx=1x\frac{d \ln{x}}{dx} = \frac{1}{x}
dln⁡udx=1ududx\frac{d \ln{u}}{dx} = \frac{1}{u}\frac{du}{dx}
dlog⁡udx=log⁡eududx\frac{d \log{u}}{dx} = \frac{\log{e}}{u}\frac{du}{dx}
dexdx=ex\frac{d\mathrm{e}^x}{dx} = \mathrm{e}^x
daxdx=axln⁡a\frac{da^x}{dx} = a^x \ln{a}
daudx=auln⁡adudx\frac{da^u}{dx} = a^u \ln{a}\frac{du}{dx}
duvdx=vuv−1dudx+uvln⁡udvdx\frac{du^v}{dx} = vu^{v-1}\frac{du}{dx} + u^v \ln{u}\frac{dv}{dx}
Expressions Containing Trigonometric Functions
dsin⁡xdx=cos⁡x or dsin⁡udx=cos⁡ududx\frac{d \sin{x}}{dx} = \cos{x} \text{ or } \frac{d \sin{u}}{dx} = \cos{u}\frac{du}{dx}
dcos⁡xdx=−sin⁡x or dcos⁡udx=−sin⁡ududx\frac{d \cos{x}}{dx} = -\sin{x} \text{ or } \frac{d \cos{u}}{dx} = -\sin{u}\frac{du}{dx}
dtan⁡xdx=sec⁡2x or dtan⁡udx=sec⁡2ududx\frac{d \tan{x}}{dx} = \sec^2{x} \text{ or } \frac{d \tan{u}}{dx} = \sec^2{u}\frac{du}{dx}
dsec⁡xdx=sec⁡xtan⁡x or dsec⁡udx=sec⁡utan⁡ududx\frac{d \sec{x}}{dx} = \sec{x}\tan{x} \text{ or } \frac{d \sec{u}}{dx} = \sec{u}\tan{u}\frac{du}{dx}
dcot⁡xdx=−csc⁡2x or dcot⁡udx=−csc⁡2ududx\frac{d \cot{x}}{dx} = -\csc^2{x} \text{ or } \frac{d \cot{u}}{dx} = -\csc^2{u}\frac{du}{dx}
dsin⁡−1xdx=11−x2 or dsin⁡−1udx=11−u2dudx\frac{d \sin^{-1}{x}}{dx} = \frac{1}{\sqrt{1-x^2}} \text{ or } \frac{d \sin^{-1}{u}}{dx} = \frac{1}{\sqrt{1-u^2}}\frac{du}{dx}
dcos⁡−1xdx=−11−x2 or dcos⁡−1udx=−11−u2dudx\frac{d \cos^{-1}{x}}{dx} = -\frac{1}{\sqrt{1-x^2}} \text{ or } \frac{d \cos^{-1}{u}}{dx} = -\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}
dtan⁡−1xdx=11+x2 or dtan⁡−1udx=11+u2dudx\frac{d \tan^{-1}{x}}{dx} = \frac{1}{1+x^2} \text{ or } \frac{d \tan^{-1}{u}}{dx} = \frac{1}{1+u^2}\frac{du}{dx}
dcot⁡−1xdx=−11+x2 or dcot⁡−1udx=−11+u2dudx\frac{d \cot^{-1}{x}}{dx} = -\frac{1}{1+x^2} \text{ or } \frac{d \cot^{-1}{u}}{dx} = -\frac{1}{1+u^2}\frac{du}{dx}

Integral Table

  • xx is any variable
  • uu is any function of xx
  • ww is dependent on uu
  • aa and bb are arbitrary constants
  • CC, the constant of integration
Fundamental Integrals
∫a dx=ax\int_{}{} a \, dx = ax
∫a f(x)dx=a∫f(x)dx\int_{}{} a \, f\left(x\right) dx = a \int_{}{} f\left(x\right) dx
∫(u+v)dx=∫u dx+∫v dx\int_{}{} \left(u+v\right) dx = \int_{}{} u\,dx + \int_{}{}v\,dx
∫u dv=uv−∫v du\int_{}{} u\,dv = uv - \int_{}{}v\,du
∫u dvdxdx=uv−∫vdudxdx \int_{}{} \frac{u\,dv}{dx} dx = uv - \int_{}{} v \frac{du}{dx} dx
∫xndx=xn+1n+1, n≠−1 \int_{}{} x^n dx = \frac{x^{n+1}}{n+1} \text{, } n \neq -1
∫x−1dx=ln⁡x\int_{}{} x^{-1} dx = \ln{x}
∫w(u)dx=∫w(u)dxduu\int_{}{} w \left( u \right) dx = \int_{}{} w\left( u \right) \frac{dx}{du} u
∫dxa2+x2=1atan⁡−1xa\int_{}{} \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a}
∫dxa2−x2=sin⁡−1xa\int_{}{} \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \frac{x}{a}
∫dxa2±x2=ln⁡(x−a2±x2)\int_{}{} \frac{dx}{\sqrt{a^2 \pm x^2}} = \ln \left( x - \sqrt{a^2 \pm x^2} \right)
∫a2−u2=12(ua2−x2+a2sin⁡−1ua)\int_{}{} \sqrt{a^2 - u^2} = \frac{1}{2} \left( u \sqrt{a^2 - x^2} + a^2 \sin^{-1} \frac{u}{a} \right)
∫dua2+u2=1atan⁡−1ua, a>0\int_{}{} \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \frac{u}{a} \text{, } a > 0
Expressions Containing Exponential and Logarithmic Functions
∫dxx=ln⁡x\int_{}{} \frac{dx}{x} = \ln{x}
∫exdx=ex\int_{}{} \mathrm{e}^{x} dx = \mathrm{e}^{x}
∫eaxdx=eaxa\int_{}{} \mathrm{e}^{ax} dx = \frac{\mathrm{e}^{ax}}{a}
∫baxdx=baxaln⁡b\int_{}{} b^{ax} dx = \frac{b^ax}{a \ln{b}}
∫ln⁡xdx=xln⁡x−x\int_{}{} \ln{x} dx = x \ln{x} - x
∫budu=buln⁡u\int_{}{} b^{u} du = \frac{b^u}{\ln{u}}
∫xeaxdx=eaxa2(ax−1)\int_{}{} x \mathrm{e}^{ax} dx = \frac{\mathrm{e}^ax}{a^2} \left( ax - 1 \right)
∫xbaxdx=xbaxaln⁡b−baxa2(ln⁡b)\int_{}{} x b^{ax} dx = \frac{x b^{ax}}{a \ln{b} } - \frac{b^{ax}}{a^2 \left( \ln{b} \right) }
∫x2eaxdx=eaxa3(a2x2−2ax+2)\int_{}{} x^2 \mathrm{e}^{ax} dx = \frac{\mathrm{e}^{ax}}{a^3} \left(a^2 x^2 - 2ax + 2 \right)
∫ln⁡axdx=xln⁡ax−x\int_{}{} \ln{ax} dx = x \ln{ax} - x
∫xln⁡axdx=x22ln⁡ax−x24\int_{}{} x \ln{ax} dx = \frac{x^2}{2} \ln{ax} - \frac{x^2}{4}
∫x2ln⁡axdx=x33ln⁡ax−x39\int_{}{} x^2 \ln{ax} dx = \frac{x^3}{3} \ln{ax} - \frac{x^3}{9}
∫(ln⁡ax)2dx=x(ln⁡ax)2−2xln⁡ax+2x\int_{}{} \left( \ln{ax} \right)^2 dx = x \left( \ln{ax} \right)^2 - 2x \ln{ax} + 2x
∫dxxln⁡ax=ln⁡(ln⁡ax)\int_{}{} \frac{dx}{x \ln{ax}} = \ln{\left( \ln{ax} \right)}
∫xnln⁡axdx=1an+1∫eydyy, where y=(n+1)ln⁡ax\int_{}{} \frac{x^n}{\ln{ax}} dx = \frac{1}{a^{n+1}} \int_{}{} \frac{\mathrm{e}^{y} dy}{y} \text{, where } y = \left(n + 1\right) \ln{ax}
Expressions Containing Trigonometric Functions
∫sin⁡xdx=−cos⁡x\int_{}{} \sin{x} dx = -\cos{x}
∫cos⁡xdx=sin⁡x\int_{}{} \cos{x} dx = \sin{x}
∫tan⁡xdx=−ln⁡(cos⁡x)\int_{}{} \tan{x} dx = -\ln{ \left( \cos{x} \right)}
∫cot⁡xdx=ln⁡(sin⁡x)\int_{}{} \cot{x} dx = \ln{\left( \sin{x} \right)}
∫sec⁡xdx=ln⁡(sec⁡x+tan⁡x)\int_{}{} \sec{x} dx = \ln{ \left( \sec{x} + \tan{x} \right)}
∫csc⁡xdx=ln⁡(csc⁡x−cot⁡x)\int_{}{} \csc{x} dx = \ln{ \left( \csc{x} - \cot{x} \right)}
∫sin⁡2udu=12u−12sin⁡ucos⁡u\int_{}{} \sin^2{u} du = \frac{1}{2} u - \frac{1}{2} \sin{u} \cos{u}
∫cos⁡2udu=12u+12sin⁡ucos⁡u\int_{}{} \cos^2{u} du = \frac{1}{2} u + \frac{1}{2} \sin{u} \cos{u}
∫csc⁡2udu=−cot⁡u\int_{}{} \csc^2{u} du = -\cot{u}
∫tan⁡2udu=tan⁡u−u\int_{}{} \tan^2{u} du = \tan{u} - u
∫cot⁡2udu=−cot⁡u−u\int_{}{} \cot^2{u} du = -\cot{u} - u
∫sin⁡axdx=−1acos⁡ax\int_{}{} \sin{ax} dx = -\frac{1}{a} \cos{ax}
∫sin⁡2axdx=x2−sin⁡2ax4a\int_{}{} \sin^2{ax} dx = \frac{x}{2}-\frac{\sin{2ax}}{4a}
∫dxsin⁡ax=1aln⁡tan⁡ax2\int_{}{} \frac{dx}{\sin{ax}} = \frac{1}{a} \ln{ \tan{\frac{ax}{2}}}
∫dxsin⁡2ax=−1acot⁡ax\int_{}{} \frac{dx}{\sin^2{ax}} = -\frac{1}{a} \cot{ax}
∫dx1±sin⁡ax=∓1atan⁡(π4∓ax2)\int_{}{} \frac{dx}{1 \pm \sin{ax}} = \mp \frac{1}{a} \tan{\left( \frac{\pi}{4} \mp \frac{ax}{2} \right) }
∫sin⁡xcos⁡xdx=12sin⁡2x\int_{}{} \sin{x} \cos{x} dx = \frac{1}{2} \sin^2{x}

Laplace Table

Time Domain f(t)=L−1{F(s)}f(t) = \mathcal{L}^{-1} \left\{ F \left( s \right) \right\}Frequency Domain F(s)=L{f(t)}F\left( s \right) = \mathcal{L} \left\{ f \left( t \right) \right\}
1 (step function) 1 \text{ (step function) }1s, where (s>0)\frac{1}{s} \text{, where } \left(s > 0\right)
tt1s2, where (s>0)\frac{1}{s^2} \text{, where } \left(s > 0\right)
tn−1t^{n-1}(n−1)!sn, where (s>0)\frac{\left( n-1 \right)!}{s^n} \text{, where } \left(s > 0\right)
t\sqrt{t}π2s32, where (s>0)\frac{{\sqrt \pi }}{{2{s^{\frac{3}{2}}}}} \text{, where } \left(s > 0\right)
1t\frac{1}{\sqrt{t}}πs12, where (s>0)\frac{{\sqrt \pi }}{{{s^{\frac{1}{2}}}}} \text{, where } \left(s > 0\right)
tn−12,     n=1,2,3,…{t^{n - \frac{1}{2}}},\,\,\,\,\,n = 1,2,3, \ldots1⋅3⋅5⋯(2n−1)π2nsn+12,where(s>0)\frac{{1 \cdot 3 \cdot 5 \cdots \left( {2n - 1} \right)\sqrt \pi }}{{{2^n}{s^{n + \frac{1}{2}}}}} {, where } \left(s > 0\right)
eat\mathrm{e}^{at}1s−a, where (s>a)\frac{1}{s-a} \text{, where } \left(s > a \right)
teatt\mathrm{e}^{at}1(s−a)2, where (s>a)\frac{1}{{\left( {s - a} \right)}^{2}} \text{, where } \left(s > a \right)
tneat,     n=1,2,3,…{t^n}{{e}^{at}},\,\,\,\,\,n = 1,2,3, \ldotsn!(s−a)n+1, where (s>a)\frac{{n!}}{{{{\left( {s - a} \right)}^{n + 1}}}} \text{, where } \left(s > a \right)
sin⁡at\sin{at} as2+a2, where (s>0)\frac{a}{s^2 + a^2} \text{, where } \left(s > 0 \right)
cos⁡at\cos{at}ss2+a2, where (s>0)\frac{s}{s^2 + a^2} \text{, where } \left(s > 0 \right)
ebtsin⁡at\mathrm{e}^{bt} \sin{at}a(s−b)2+a2, where (s>b)\frac{a}{\left(s - b \right)^2 + a^2} \text{, where } \left(s > b \right)
ebtcos⁡at\mathrm{e}^{bt} \cos{at}s−b(s−b)2+a2, where (s>b)\frac{s-b}{\left(s - b \right)^2 + a^2} \text{, where } \left(s > b \right)
tsin⁡att \sin{at}2as(s2−a2)2, where (s>a)\frac{2as}{\left(s^2 - a^2\right)^2} \text{, where } \left(s > a \right)
tcos⁡att \cos{at}s2−a2(s2+a2)2, where (s>0)\frac{s^2 - a^2}{\left(s^2 + a^2\right)^2} \text{, where } \left(s > 0 \right)
sinh⁡at\sinh{at}a(s2−a2)2, where (s>∣a∣)\frac{a}{\left(s^2 - a^2\right)^2} \text{, where } \left(s > \lvert a \rvert \right)
cosh⁡at\cosh{at}s(s2−a2)2, where (s>∣a∣)\frac{s}{\left(s^2 - a^2\right)^2} \text{, where } \left(s > \lvert a \rvert \right)
sin⁡(at+b)\sin \left(at + b\right)ssin⁡b+acos⁡b(s2+a2)2\frac{s\sin{b} + a \cos{b}}{\left(s^2 + a^2\right)^2}
cos⁡(at+b)\cos \left(at + b\right)scos⁡b−asin⁡b(s2+a2)2\frac{s\cos{b} - a \sin{b}}{\left(s^2 + a^2\right)^2}
eat−ebta−b\frac{\mathrm{e}^{at} - \mathrm{e}^{bt}}{a - b}1(s−a)(s−b)\frac{1}{\left(s-a\right)\left(s-b\right)}
aeat−bebta−b\frac{a\mathrm{e}^{at} - b\mathrm{e}^{bt}}{a - b}s(s−a)(s−b)\frac{s}{\left(s-a\right)\left(s-b\right)}
δ (impulse function)\delta \text{ (impulse function)}11
square wave, period =2c\text{square wave, period }=2c1s(1+e−cs)\frac{1}{s \left( 1 + \mathrm{e}^{-cs} \right)}
triangular wave, period =2c\text{triangular wave, period }=2c1−e−css2(1+e−cs)\frac{1 - \mathrm{e}^{-cs}}{s^2 \left( 1 + \mathrm{e}^{-cs} \right)}
sin⁡atsin⁡bt\sin{at} \sin{bt} 2abs(s2+(a+b)2)(s2+(a−b)2)\frac{2abs}{\left( s^2 + \left(a+b\right)^2\right) \left( s^2 + \left(a-b\right)^2\right)}
1−cos⁡ata2 \frac{1 - \cos{at}}{a^2}  1s(s2+a2)\frac{1}{s \left( s^2 + a^2 \right) }
at−sin⁡ata3 \frac{at - \sin{at}}{a^3}  1s2(s2+a2)\frac{1}{s^2 \left( s^2 + a^2 \right) }
sin⁡at−atcos⁡at2a3 \frac{\sin{at} - at\cos{at}}{2a^3}  1(s2+a2)2\frac{1}{\left( s^2 + a^2 \right)^2 }
AlgebraGeometry