Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Math formulas: Most popular formulas

0 formulas included in custom cheat sheet

Here is a list of math formulas that have been downloaded most number of times.

a2b2=(ab)(a+b) a^2 - b^2 = (a-b)(a+b)
a3b3=(ab)(a2+ab+b2) a^3 - b^3 = (a-b)\left(a^2 + ab + b^2\right)
a3+b3=(a+b)(a2ab+b2) a^3 + b^3 = (a+b)\left(a^2 - ab + b^2\right)
(a+b)2=a2+2ab+b2 (a + b)^2 = a^2 + 2ab + b^2
(ab)2=a22ab+b2 (a - b)^2 = a^2 - 2ab + b^2
x1,2=b±b24ac2a x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
(ab)3=a33a2b+3ab2b3 (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
AB={x:xA  or  xB} A \cup B = \left\{x : x \in A ~~ or ~~ x \in B \right\}
AB={x:xA  and  xB} A \cap B = \left\{x : x \in A ~~ and ~~ x \in B \right\}
(a+b)3=a3+3a2b+3ab2+b3 (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
secα=1cosα=HypotenuseAdjacent \sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}
sin(α+β)=sinαcosβ+sinβcosα \sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha
cos(α+β)=cosαcosβsinαcosβ \cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta
cotα=1tanα=AdjacentOpposite \cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}
sinα=OppositeHypotenuse \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}
cscα=1sinα=HypotenuseOpposite \csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}
cos(2α)=cos2αsin2α \cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha
sin(2α)=2sinαcosα \sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha
sin2x+cos2x=1 \sin^2x + \cos^2x = 1
tan2x+1=1cos2x \tan^2x + 1 = \frac{1}{\cos^2x}
y=mx+b y = mx+b
C=πd=2πr C = \pi \cdot d = 2\cdot \pi \cdot r
A=r2π A = r^2\pi
y2=2px y^2 = 2\,p\,x
DC2=DGDE DC^2 = DG \cdot DE
DHDG=DFDE DH \cdot DG = DF \cdot DE
if aby=bax and y=baxif a<by=abx and y=abx \begin{aligned} & \text{if } a \geq b \Longrightarrow y = \frac{b}{a}x \text{ and } y = -\frac{b}{a}x \\ & \text{if } a < b \Longrightarrow y = \frac{a}{b}x \text{ and } y = -\frac{a}{b}x \\ \end{aligned}
if abF1(a2+b2,0)  F2(a2+b2,0)if a<bF1(0,a2+b2)  F2(0,a2+b2) \begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2+b^2},0\right)~~ F_2\left(\sqrt{a^2+b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{a^2+b^2}\right) ~~ F_2\left(0, \sqrt{a^2+b^2}\right) \end{aligned}
x=asinty=bsintcost \begin{aligned} x &= \frac{a}{\sin t} \\ y &= \frac{b\,\sin t}{\cos t} \end{aligned}
x2a2y2b2=1 \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
ddx(logax)=1xlna,x>0 \frac{d}{dx} \left( \log_a x \right) = \frac{1}{x\cdot \ln a} , x > 0
(fg)=fg+fg (f \cdot g)' = f' \cdot g + f \cdot g'
ddx(ex)=ex \frac{d}{dx} (e^x) = e^x
(fg)=fgfgg2 \left( \frac{f}{g} \right)' = \frac{ f'\cdot g - f \cdot g' }{g^2}
(f(g(x)))=f(g(x))g(x) \left( f \left(g(x) \right) \right)' = f'(g(x)) \cdot g'(x)
ddx(ax)=axlna \frac{d}{dx} (a^x) = a^x \cdot \ln a
ddx(lnx)=1x,x>0 \frac{d}{dx} (\ln x) = \frac{1}{x} , x > 0
ddx(arctanx)=11+x2 \frac{d}{dx} (\arctan x) = \frac{1}{1+x^2}
ddx(cscx)=cscxcotx \frac{d}{dx} (\csc x) = - \csc x \cdot \cot x
ddx(cotx)=1sin2x \frac{d}{dx} (\cot x) = -\frac{1}{ \sin^2x }
ecxxn=1n1(ecxxn1+cecxxn1dx) \int \frac{e^{cx}}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}} + c\cdot \int \frac{e^{cx}}{x^{n-1}} dx \right)
xecx=ecxc2(cx1) \int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1)
ecxdx=1cecx \int e^{cx}dx = \frac{1}{c}e^{cx}
dxxlnx=lnlnx \int \frac{dx}{x\cdot \ln x} = \ln|\ln x|
lnxnxdx=(lnxn)22n,(for n0) \int \frac{\ln x^n}{x}dx = \frac{\left(\ln x^n \right)^2}{2n},\quad (\text{for } n \ne 0 )
(lnx)2dx=x(lnx)22xlnx+2x \int (\ln x)^2dx = x(\ln x)^2 - 2x\ln x + 2x
(ln(cx))ndx=x(lnx)nn(ln(cx))n1dx \int (\ln (cx))^ndx = x(\ln x)^n - n\cdot\int (\ln (cx))^{n-1}dx
dxlnx=lnlnx+lnx+n=2(lnx)iii! \int \frac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\frac{(\ln x)^i}{i\cdot i!}
dx(lnx)n=x(n1)(lnx)n1+1n1dx(lnx)n1 \int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1} \int \frac{dx}{(\ln x)^{n-1}}
xmlnxdx=xm+1(lnxm+11(m+1)2)(fot m1) \int x^m \cdot \ln xdx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2} \right) \quad ( \text{fot } m\ne1)
0dxx2+a2=π2a \int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}
0xmxn+an=πam+1nnsin[(m+1)π/n], 0<m+1<n \int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n
0adxa2x2=π2 \int^a_0 \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}
0aa2x2dx=πa24 \int^a_0 \sqrt{a^2 - x^2}\,dx = \frac{\pi\,a^2}{4}
0axm(anxn)pdx=am+1+np Γ[(m+1)/n] Γ(p+1)nΓ[(m+1)/n+p+1] \int^a_0 x^m \left(a^n - x^n\right)^p\,dx = \frac{a^{m+1+np}~\Gamma[(m+1)/n]~\Gamma(p+1) }{n\,\Gamma[(m+1)/n + p +1]}
0xp1dx1+x=πsin(pπ), 0<p<1 \int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1
0xsin(mx)x2+a2dx=π2ema \int^\infty_0 \frac{x\,\sin (mx)}{x^2 + a^2} dx = \frac{\pi}{2}e^{-ma}
0πcos(mx)cos(nx)dx={0m,n integers and mnπ/2m,n integers and m=n \int^\pi_0 \cos (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m\ne n \\ \pi/2 & \quad m , n \text{ integers and } m = n \end{array} \right.
0πsin(mx)cos(nx)dx={0m,n integers and m+n odd2m/(m2n2)m,n integers and m+n even \int^\pi_0 \sin (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m + n \text{ odd} \\ 2m/(m^2 - n^2) & \quad m , n \text{ integers and } m + n \text{ even} \end{array} \right.
0π/2sin2mxdx=0π/2cos2mxdx=1352m12462mπ2 \int^{\pi/2}_0 \sin^{2m}x\,dx = \int^{\pi/2}_0 \cos^{2m}x\,dx = \frac{1\cdot3\cdot5\dots 2m-1}{2\cdot 4 \cdot 6 \dots 2m} \frac{\pi}{2}
ex=1+x+x22!+x33!+ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
f(x)=f(a)+f(a)(xa)+f(a)(xa)22!++f(n1)(a)(xa)n1(n1)!+Rn f(x) = f(a) + f'(a)(x-a) + \frac{f{'}{'}(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n
S=a11q,(for 1<q<1) S = \frac{a_1}{1-q}, \quad (\text{for } -1 < q < 1)
ln(1+x)=xx22+x33x44+1<x1 \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad -1 < x \leq 1
sinx=xx33!+x55!x77!+ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
cosx=1x22!+x44!x66!+ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
(a+x)n=an+nan1+n(n1)2!an2x2+n(n1)(n2)3!an3x3+=an+(n1)an1x+(n2)an2x2+(n3)an3x3+ \begin{aligned} (a + x)^n &= a^n + na^{n-1} + \frac{n(n-1)}{2!} a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots \\ &= a^n + { n \choose 1} a^{n-1}x + { n \choose 2} a^{n-2}x^2 + { n \choose 3} a^{n-3}x^3 + \cdots \end{aligned}
cotx=1xx3x34522nBnx2n1(2n)!0<x<π \cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi
secx=1+x22+5x424+61x6720++Enx2n(2n)!π2<x<π2 \sec x = 1+ \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots + \frac{E_n x^{2n}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}
cscx=1x+x6+7x3360++2(22n1)Enx2n(2n)!0<x<π \csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots + \frac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi

Was these formulas helpful?

Yes No