math_formulaAlgebra
 ((a+b))2 = a2 + b2 + 2 a b$\left(a+b{\right)}^{2}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}{a}^{2}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}{b}^{2}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}2\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}b$ ((a-b))2 = a2 + b2 - 2 a b$\left(a-b{\right)}^{2}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}{a}^{2}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}{b}^{2}\phantom{\rule{0.22em}{0ex}}-\phantom{\rule{0.22em}{0ex}}2\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}b$ ((a+b))3 = a3 + 3a2b+3ab2+b3 $\left(a+b{\right)}^{3}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}{a}^{3}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}3{a}^{2}b+3a{b}^{2}+{b}^{3}\phantom{\rule{0.22em}{0ex}}$ ((a-b))3 = a3 - 3a2b+3ab2-b3 $\left(a-b{\right)}^{3}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}{a}^{3}\phantom{\rule{0.22em}{0ex}}-\phantom{\rule{0.22em}{0ex}}3{a}^{2}b+3a{b}^{2}-{b}^{3}\phantom{\rule{0.22em}{0ex}}$ a3+b3=((a+b))(a2-ab+b2)${a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)$ a3-b3=((a-b))(a2+ab+b2)${a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)$ ex=1+x1!+x22!+x33!+x44!+x55!+ . . . . . ${e}^{x}=1+\frac{x}{1!}+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+\frac{{x}^{5}}{5!}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}$ ln((1+x))=x1-x22+x33-x44+x55-x66+ . .$ln\left(1+x\right)=\frac{x}{1}-\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}-\frac{{x}^{4}}{4}+\frac{{x}^{5}}{5}-\frac{{x}^{6}}{6}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.$ cos x=1-x22!+x44!-x66!+x88! - . . . .$cos\phantom{\rule{0.22em}{0ex}}x=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\frac{{x}^{6}}{6!}+\frac{{x}^{8}}{8!}\phantom{\rule{0.22em}{0ex}}-\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.$
Complex Number
$z=a+ib=r{e}^{i\mathrm{\pi }}\phantom{\rule{0.22em}{0ex}}$
$i=\sqrt{-1}$
$\stackrel{Β―}{z}=a-ib$
$|z|=\sqrt{{a}^{2}+{b}^{2}}$
$|z|=|-z|$
$z\phantom{\rule{0.22em}{0ex}}\stackrel{Β―}{\phantom{\rule{0.22em}{0ex}}z\phantom{\rule{0.22em}{0ex}}}\phantom{\rule{0.22em}{0ex}}=|z{|}^{2}\phantom{\rule{0.22em}{0ex}}$
$|{z}_{1}{z}_{2}|=|{z}_{1}|\phantom{\rule{0.22em}{0ex}}|{z}_{2}|\phantom{\rule{0.22em}{0ex}}$
$\phantom{\rule{0.22em}{0ex}}|\begin{array}{c}\frac{{z}_{1}}{{z}_{2}}\end{array}|=\frac{|\begin{array}{c}{z}_{1}\end{array}|}{|\begin{array}{c}{z}_{2}\end{array}|}$
$\left(cos\mathrm{\pi }+isin\mathrm{\pi }{\right)}^{n}=cos\phantom{\rule{0.22em}{0ex}}n\mathrm{\pi }+i\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}n\mathrm{\pi }\phantom{\rule{0.22em}{0ex}}$
$amp\left(z\right)=arg\left(z\right)=\mathrm{\pi }=ta{n}^{-1}\frac{b}{a}\phantom{\rule{0.22em}{0ex}}$
Cube root of unity$=1,\phantom{\rule{0.22em}{0ex}}\mathrm{\pi },\phantom{\rule{0.22em}{0ex}}{\mathrm{\pi }}^{2}\phantom{\rule{0.22em}{0ex}}$$\mathrm{\pi }=\left(\frac{-1+i\sqrt{3}}{2}\right)$${\mathrm{\pi }}^{2}=\left(\frac{-1-i\sqrt{3}}{2}\right)$
Coordinate geometry
 Point, Line Distance formulaL=((x1-x2))2+((y1-y2))2$L=\sqrt{\left({x}_{1}-{x}_{2}{\right)}^{2}+\left({y}_{1}-{y}_{2}{\right)}^{2}}$ Slope of a line when coordinates of any two points on the line are givenslope=m=y2-y1x2-x1$slope=m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ are parallel if m1=m2${m}_{1}={m}_{2}$ Two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ are perpendicular if m1.m2=-1${m}_{1}.{m}_{2}=-1$ Angle between two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ tanπ=m2-m11+m1m2$tan\mathrm{\pi }=\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}$ Equation of straight line: Point slope formulam=y-y0x-x0$m=\frac{y-{y}_{0}}{x-{x}_{0}}$ Equation of straight line: Point slope formulam=y-y0x-x0$m=\frac{y-{y}_{0}}{x-{x}_{0}}$ Equation of straight line: two point formulay-y1x-x1=y2-y1x2-x1$\frac{y-{y}_{1}}{x-{x}_{1}}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Equation of straight line: slope intercept formy=mx+c$y=mx+c$where c is y-intercept Equation of straight line: intercept formxa+yb=1$\frac{x}{a}+\frac{y}{b}=1$where, a is x-intercept and b is y-intercept Equation of straight line: normal formx cosπ+y sin π=p $x\phantom{\rule{0.22em}{0ex}}cos\mathrm{\pi }+y\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\mathrm{\pi }=p\phantom{\rule{0.22em}{0ex}}$where, p is normal distance of line from origin and π $\mathrm{\pi }\phantom{\rule{0.22em}{0ex}}$ is angle which normal makes with positive x-axis. Conic Section Equation of circle with center (h,k) and radius r will be((x-h))2+((y-k))2=r2$\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$
Set Theory
 A binary operation * on the set X is called commutative, if a β b = b β a$a\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}b\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}b\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}a$ for every a, bβX$a,\phantom{\rule{0.22em}{0ex}}b\beta X$ Given a binary operation β : A Γ A βA$\beta \phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}\Gamma \phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}\beta A$ , an element e β A$e\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}A$ , if exists, if a β e = a = e β a, β a β A$a\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}e\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}e\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}a,\phantom{\rule{1.1em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}A$ Given a binary operation β : A Γ A βA$\beta \phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}\Gamma \phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}\beta A$ , with the identity element e in A , an element a β A$a\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}A$ is said to be invertible with respect to the operation * , if there exist an element b in A such that a β b = e = b β a $a\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}b\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}e\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}b\phantom{\rule{0.22em}{0ex}}\beta \phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.66em}{0ex}}$then b is called inverse of a and is denoted by a-1${a}^{-1}$
Matrix and Determinants
 (AT)T=A$\left({A}^{T}{\right)}^{T}=A$ ((kA))T=kAT$\left(kA{\right)}^{T}=k{A}^{T}$ ((A+B))T=AT+BT$\left(A+B{\right)}^{T}={A}^{T}+{B}^{T}$ ((AB))T=BTAT$\left(AB{\right)}^{T}={B}^{T}{A}^{T}$ A square matrix A = [[aij]] $A\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\left[{a}_{ij}\right]\phantom{\rule{0.22em}{0ex}}$ is said to be symmetric if AT= A i.e. [[aij]] = [[aji]] ${A}^{T}=\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{1.32em}{0ex}}i.e.\phantom{\rule{0.66em}{0ex}}\left[{a}_{ij}\right]\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\left[{a}_{ji}\right]\phantom{\rule{0.44em}{0ex}}$ for all possible values of i$i$ and j$j$ A square matrix A = [[aij]] $A\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\left[{a}_{ij}\right]\phantom{\rule{0.22em}{0ex}}$ is said to be skew symmetric if AT= -A i.e. [[aij]] =-[[aji]] ${A}^{T}=\phantom{\rule{0.22em}{0ex}}-A\phantom{\rule{1.32em}{0ex}}i.e.\phantom{\rule{0.66em}{0ex}}\left[{a}_{ij}\right]\phantom{\rule{0.22em}{0ex}}=-\left[{a}_{ji}\right]\phantom{\rule{0.44em}{0ex}}$ for all possible values of i$i$ and j$j$ . So, diagonal elements of a skew symmetric matrix are zero. For any square matrix [A] with real number entries,A + AT $A\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}{A}^{T}\phantom{\rule{0.22em}{0ex}}$ is symmetric matrixA β AT$A\phantom{\rule{0.22em}{0ex}}\mathrm{\beta }\phantom{\rule{0.22em}{0ex}}{A}^{T}$ is a skew symmetric matrix
Permuntation and Combination
 nCr=n!((n-r))! r!${}^{n}{C}_{r}=\frac{n!}{\left(n-r\right)!\phantom{\rule{0.22em}{0ex}}r!}$ nCr=nCn-r${}^{n}{C}_{r}{=}^{n}{C}_{n-r}$ nPr${}^{n}{P}_{r}$= r! nCr$\phantom{\rule{0.22em}{0ex}}r!{\phantom{\rule{0.22em}{0ex}}}^{n}{C}_{r}$ nCr+nCr-1=n+1Cr${}^{n}{C}_{r}{+}^{n}{C}_{r-1}{=}^{n+1}{C}_{r}$ nC1+nC2+nC3+ . . . +nCn=2n ${}^{n}{C}_{1}{+}^{n}{C}_{2}{+}^{n}{C}_{3}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}{+}^{n}{C}_{n}={2}^{n}\phantom{\rule{0.22em}{0ex}}$ nC0+nC2+. . =nC1+nC3 . . . =2n-1 ${}^{n}{C}_{0}{+}^{n}{C}_{2}+.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}{=}^{n}{C}_{1}{+}^{n}{C}_{3}\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}={2}^{n-1}\phantom{\rule{0.22em}{0ex}}$ nPr=n!((n-r))!${}^{n}{P}_{r}=\frac{n!}{\left(n-r\right)!}$ nPn=n!${}^{n}{P}_{n}=n!$ nPr=nΓn-1Pr-1${}^{n}{P}_{r}=n{\Gamma }^{n-1}{P}_{r-1}$
calculus Calculus Continuity
Suppose f is a real function on a subset of real numbers then c be a point in the domain of "f", then "f" is continupos at "c" if : $\begin{array}{c}Lim\\ x\beta c\end{array}f\left(x\right)\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}f\left(c\right)\phantom{\rule{0.22em}{0ex}}$
If "f" and "g" are two real functions continuous at a real number "c", then (a) $f+g\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}continuous\phantom{\rule{0.22em}{0ex}}at\phantom{\rule{0.22em}{0ex}}x=c$(b) $f-g\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}continuous\phantom{\rule{0.22em}{0ex}}at\phantom{\rule{0.22em}{0ex}}x=c$(c) $f.g\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}continuous\phantom{\rule{0.22em}{0ex}}at\phantom{\rule{0.22em}{0ex}}x=c$(d) $\left(\frac{f}{g}\right)\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}continuous\phantom{\rule{0.22em}{0ex}}at\phantom{\rule{0.22em}{0ex}}x=c$
Limits
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{sinx}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{tanx}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1$
$\begin{array}{c}Lim\\ x\beta a\end{array}\frac{{x}^{n}-{a}^{n}}{x-a}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}n\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}{x}^{n-1}$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{{e}^{x}-1}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\left(1+x{\right)}^{\frac{1}{x}}=e$
$\begin{array}{c}Lim\\ x\beta \mathrm{\beta }\end{array}\left(1+\frac{a}{x}{\right)}^{x}={e}^{a}$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{1-cosx}{x}=0$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{si{n}^{-1}x}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{{a}^{x}-1}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}ln\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}lo{g}_{e}\phantom{\rule{0.22em}{0ex}}a$
$\begin{array}{c}Lim\\ x\beta 0\end{array}\frac{ln\left(1+x\right)}{x}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1$
$\begin{array}{c}Lim\\ x\beta \mathrm{\beta }\end{array}\left(1+\frac{1}{x}{\right)}^{x}=e$
if $\phantom{\rule{0.22em}{0ex}}\begin{array}{c}Lim\\ x\beta a\end{array}\frac{f\left(x\right)}{g\left(x\right)}\phantom{\rule{0.44em}{0ex}}gives\phantom{\rule{0.22em}{0ex}}form\phantom{\rule{0.44em}{0ex}}\frac{0}{0}\phantom{\rule{0.44em}{0ex}}$ as $\phantom{\rule{0.22em}{0ex}}f\left(a\right)=0\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}g\left(a\right)=0$ then $\begin{array}{c}Lim\\ x\beta a\end{array}\frac{f\left(x\right)}{g\left(x\right)}\phantom{\rule{0.44em}{0ex}}=\frac{f\text{'}\left(a\right)}{g\text{'}\left(a\right)}\phantom{\rule{0.22em}{0ex}}$
Differentiation
 ddx((uv))=ududx+vdvdx $\frac{d}{dx}\left(uv\right)=u\frac{du}{dx}+v\frac{dv}{dx}\phantom{\rule{0.22em}{0ex}}$ ddx(xn)=nxn-1 $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}\phantom{\rule{0.44em}{0ex}}$ ddx((sin x))=cos x $\frac{d}{dx}\left(sin\phantom{\rule{0.22em}{0ex}}x\right)=cos\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.88em}{0ex}}$ ddx((cosx))=-sin x $\frac{d}{dx}\left(cosx\right)=-sin\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}$ ddx((tan x))=sec2x $\frac{d}{dx}\left(tan\phantom{\rule{0.22em}{0ex}}x\right)=se{c}^{2}x\phantom{\rule{0.22em}{0ex}}$ ddx((cosec x))=-cosec x.cot x$\frac{d}{dx}\left(cosec\phantom{\rule{0.22em}{0ex}}x\right)=-cosec\phantom{\rule{0.22em}{0ex}}x.cot\phantom{\rule{0.22em}{0ex}}x$ ddx((sec x))=sec x.tan x$\frac{d}{dx}\left(sec\phantom{\rule{0.22em}{0ex}}x\right)=sec\phantom{\rule{0.22em}{0ex}}x.tan\phantom{\rule{0.22em}{0ex}}x$ ddx((cot x))=-cosec2x$\frac{d}{dx}\left(cot\phantom{\rule{0.22em}{0ex}}x\right)=-cose{c}^{2}x$ ddx((ln x))=1x$\frac{d}{dx}\left(ln\phantom{\rule{0.22em}{0ex}}x\right)=\frac{1}{x}$ ddx(ex)=ex$\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$ ddx(ax)=ax ln a $\frac{d}{dx}\left({a}^{x}\right)={a}^{x}\phantom{\rule{0.22em}{0ex}}ln\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.88em}{0ex}}$ ddx(sin-1x)=11-x2 $\frac{d}{dx}\left(si{n}^{-1}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}\phantom{\rule{2.42em}{0ex}}$ ddx((logax))=1x ln a $\phantom{\rule{0.22em}{0ex}}\frac{d}{dx}\left(lo{g}_{a}x\right)=\frac{1}{x\phantom{\rule{0.22em}{0ex}}ln\phantom{\rule{0.22em}{0ex}}a}\phantom{\rule{0.88em}{0ex}}$ ddx(cos-1x)=-11-x2$\frac{d}{dx}\left(co{s}^{-1}x\right)=-\frac{1}{\sqrt{1-{x}^{2}}}$ ddx(tan-1x)=11+x2 $\frac{d}{dx}\left(ta{n}^{-1}x\right)=\frac{1}{1+{x}^{2}}\phantom{\rule{3.3em}{0ex}}$ ddx(cot-1x)=-11+x2 $\frac{d}{dx}\left(co{t}^{-1}x\right)=-\frac{1}{1+{x}^{2}}\phantom{\rule{3.52em}{0ex}}$ ddx(sec-1x)=1xx2-1$\frac{d}{dx}\left(se{c}^{-1}x\right)=\frac{1}{x\sqrt{{x}^{2}-1}}$ ddx(cosec-1x)=-1xx2-1$\frac{d}{dx}\left(cose{c}^{-1}x\right)=-\frac{1}{x\sqrt{{x}^{2}-1}}$
Integration
 β«xn dx=xn+1n+1 + C$\phantom{\rule{0.22em}{0ex}}\beta «{x}^{n}\phantom{\rule{0.22em}{0ex}}dx=\frac{{x}^{n+1}}{n+1}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«cos x dx=sin x + C$\beta «cos\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=sin\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«sin x dx=-cos x + C$\phantom{\rule{0.22em}{0ex}}\beta «sin\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=-cos\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«sec2x dx=tan x + C$\beta «se{c}^{2}x\phantom{\rule{0.22em}{0ex}}dx=tan\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«cosec x.cot x dx=-cosec x + C$\beta «cosec\phantom{\rule{0.22em}{0ex}}x.cot\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=-cosec\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«sec x.tan x dx=sec x + C$\beta «sec\phantom{\rule{0.22em}{0ex}}x.tan\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=sec\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«cosec2 x dx=-cot x + C$\beta «cose{c}^{2}\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=-cot\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«dxx=ln x+ C $\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{x}=ln\phantom{\rule{0.22em}{0ex}}x+\phantom{\rule{0.22em}{0ex}}C\phantom{\rule{0.22em}{0ex}}$ β«ex dx=ex + C $\phantom{\rule{0.44em}{0ex}}\beta «{e}^{x}\phantom{\rule{0.22em}{0ex}}dx={e}^{x}\phantom{\rule{0.44em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C\phantom{\rule{0.22em}{0ex}}$ β«ax dx=axln a + C $\phantom{\rule{0.44em}{0ex}}\beta «{a}^{x}\phantom{\rule{0.22em}{0ex}}dx=\frac{{a}^{x}}{ln\phantom{\rule{0.22em}{0ex}}a}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C\phantom{\rule{0.22em}{0ex}}$ β«dx1-x2=sin-1x+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{\sqrt{1-{x}^{2}}}=si{n}^{-1}x+\phantom{\rule{0.22em}{0ex}}C$ β«dxx1-x2=cosec-1x+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{x\sqrt{1-{x}^{2}}}=cose{c}^{-1}x+\phantom{\rule{0.22em}{0ex}}C$ β«dxx1-x2=-cosec-1x+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{x\sqrt{1-{x}^{2}}}=-cose{c}^{-1}x+\phantom{\rule{0.22em}{0ex}}C$ β«dxx2-a2=12alog|x-ax+a|+ C$\beta «\frac{dx}{{x}^{2}-{a}^{2}}=\frac{1}{2a}log|\frac{x-a}{x+a}|+\phantom{\rule{0.22em}{0ex}}C$ β«dxa2-x2=12alog|a+xa-x|+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}log|\frac{a+x}{a-x}|+\phantom{\rule{0.22em}{0ex}}C$ β«dxx2+a2=1atan-1 x a+ C$\phantom{\rule{0.44em}{0ex}}\beta «\frac{dx}{{x}^{2}+{a}^{2}}=\frac{1}{a}ta{n}^{-1}\frac{\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}}{\phantom{\rule{0.22em}{0ex}}a}+\phantom{\rule{0.22em}{0ex}}C$ β«dxx2-a2=log|x+x2-a2|+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{\sqrt{{x}^{2}-{a}^{2}}}=log|x+\sqrt{{x}^{2}-{a}^{2}}|+\phantom{\rule{0.22em}{0ex}}C$ β«dxa2-x2=sin-1 x a + C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}=si{n}^{-1}\frac{\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}}{\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}}+\phantom{\rule{0.22em}{0ex}}C$ β«dxx2+a2=log|x+x2+a2|+ C$\phantom{\rule{0.22em}{0ex}}\beta «\frac{dx}{\sqrt{{x}^{2}+{a}^{2}}}=log|x+\sqrt{{x}^{2}+{a}^{2}}|+\phantom{\rule{0.22em}{0ex}}C$ β«sec x dx=log |sec x+tan x| + C$\beta «sec\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=log\phantom{\rule{0.22em}{0ex}}|sec\phantom{\rule{0.22em}{0ex}}x+tan\phantom{\rule{0.22em}{0ex}}x|\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«cosec x dx=log |cosec x-cot x| + C$\beta «cosec\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=log\phantom{\rule{0.22em}{0ex}}|cosec\phantom{\rule{0.22em}{0ex}}x-cot\phantom{\rule{0.22em}{0ex}}x|\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«cot x dx=log |sin x| + C$\beta «cot\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=log\phantom{\rule{0.22em}{0ex}}|sin\phantom{\rule{0.22em}{0ex}}x|\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«tan x dx=log |sec x| + C$\beta «tan\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}dx=log\phantom{\rule{0.22em}{0ex}}|sec\phantom{\rule{0.22em}{0ex}}x|\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C$ β«x2+a2 dx=12xx2+a2 + a22log|x+x2+a2|+ C$\phantom{\rule{0.22em}{0ex}}\beta «\sqrt{{x}^{2}+{a}^{2}}\phantom{\rule{0.22em}{0ex}}dx=\frac{1}{2}x\sqrt{{x}^{2}+{a}^{2}}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}\frac{{a}^{2}}{2}log|x+\sqrt{{x}^{2}+{a}^{2}}|+\phantom{\rule{0.22em}{0ex}}C$ β«x2-a2 dx=12xx2-a2 - a22log|x+x2-a2|+ C$\phantom{\rule{0.22em}{0ex}}\beta «\sqrt{{x}^{2}-{a}^{2}}\phantom{\rule{0.22em}{0ex}}dx=\frac{1}{2}x\sqrt{{x}^{2}-{a}^{2}}\phantom{\rule{0.22em}{0ex}}-\phantom{\rule{0.22em}{0ex}}\frac{{a}^{2}}{2}log|x+\sqrt{{x}^{2}-{a}^{2}}|+\phantom{\rule{0.22em}{0ex}}C$ β«a2-x2 dx=12xa2-x2 + a22sin-1xa+ C$\phantom{\rule{0.22em}{0ex}}\beta «\sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{0.22em}{0ex}}dx=\frac{1}{2}x\sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}\frac{{a}^{2}}{2}si{n}^{-1}\frac{x}{a}+\phantom{\rule{0.22em}{0ex}}C$ β«f((x))g((x))=f((x))β«g((x))dx-β«[f'((x))β«g((x))dx]dx$\beta «f\left(x\right)g\left(x\right)=f\left(x\right)\beta «g\left(x\right)dx-\beta «\left[f\text{'}\left(x\right)\beta «g\left(x\right)dx\right]dx$ β«ex[[f((x))+f'((x))]] dx=exf((x)) + C $\beta «{e}^{x}\left[f\left(x\right)+f\text{'}\left(x\right)\right]\phantom{\rule{0.22em}{0ex}}dx={e}^{x}f\left(x\right)\phantom{\rule{0.22em}{0ex}}+\phantom{\rule{0.22em}{0ex}}C\phantom{\rule{0.22em}{0ex}}$
trigonometry Trigonometry
 π radian = 180Β°$\mathrm{\pi }\phantom{\rule{0.22em}{0ex}}radian\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}180\mathrm{Β°}$ sin((-A))=-sinA$sin\left(-A\right)=-sinA$ cosec((-A))=-cosecA$cosec\left(-A\right)=-cosecA$ cos((-A))=cosA$cos\left(-A\right)=cosA$ sec((-A))=secA$sec\left(-A\right)=secA$ tan((-A))=-tanA$tan\left(-A\right)=-tanA$ cot((-A))=-cotA$cot\left(-A\right)=-cotA$ sin(π2-A)=cos A$sin\left(\frac{\mathrm{\pi }}{2}-A\right)=cos\phantom{\rule{0.22em}{0ex}}A$ cos(π2-A)=sin A$cos\left(\frac{\mathrm{\pi }}{2}-A\right)=sin\phantom{\rule{0.22em}{0ex}}A$ tan(π2-A)=cot A$tan\left(\frac{\mathrm{\pi }}{2}-A\right)=cot\phantom{\rule{0.22em}{0ex}}A$ cosec(π2-A)=sec A$cosec\left(\frac{\mathrm{\pi }}{2}-A\right)=sec\phantom{\rule{0.22em}{0ex}}A$ sec(π2-A)=cosec A$\phantom{\rule{0.22em}{0ex}}sec\left(\frac{\mathrm{\pi }}{2}-A\right)=cosec\phantom{\rule{0.22em}{0ex}}A$ cot(π2-A)=tan A$cot\left(\frac{\mathrm{\pi }}{2}-A\right)=tan\phantom{\rule{0.22em}{0ex}}A$ sin(π2+A)=cos A $sin\left(\frac{\mathrm{\pi }}{2}+A\right)=cos\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.66em}{0ex}}$ cos(π2+A)=-sin A $cos\left(\frac{\mathrm{\pi }}{2}+A\right)=-sin\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.44em}{0ex}}$ tan(π2+A)=-cot A$tan\left(\frac{\mathrm{\pi }}{2}+A\right)=-cot\phantom{\rule{0.22em}{0ex}}A$ cosec(π2+A)=sec A $\phantom{\rule{0.22em}{0ex}}cosec\left(\frac{\mathrm{\pi }}{2}+A\right)=sec\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}$ sec(π2+A)=-cosec A$\phantom{\rule{0.22em}{0ex}}sec\left(\frac{\mathrm{\pi }}{2}+A\right)=-cosec\phantom{\rule{0.22em}{0ex}}A$ cot(π2+A)=-tan A$\phantom{\rule{0.22em}{0ex}}cot\left(\frac{\mathrm{\pi }}{2}+A\right)=-tan\phantom{\rule{0.22em}{0ex}}A$ sin((π-A))=sin A $sin\left(\mathrm{\pi }-A\right)=sin\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.88em}{0ex}}$ cos((π-A))=-cos A $cos\left(\mathrm{\pi }-A\right)=-cos\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}$ tan((π-A))=-tan A $tan\left(\mathrm{\pi }-A\right)=-tan\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.66em}{0ex}}$ cosec((π-A))=cosec A$cosec\left(\mathrm{\pi }-A\right)=cosec\phantom{\rule{0.22em}{0ex}}A$ sec((π-A))=-sec A$\phantom{\rule{0.22em}{0ex}}sec\left(\mathrm{\pi }-A\right)=-sec\phantom{\rule{0.22em}{0ex}}A$ cot((π-A))=-cot A$cot\left(\mathrm{\pi }-A\right)=-cot\phantom{\rule{0.22em}{0ex}}A$ sin((π+A))=-sin A$sin\left(\mathrm{\pi }+A\right)=-sin\phantom{\rule{0.22em}{0ex}}A$ cos((π+A))=-cos A $cos\left(\mathrm{\pi }+A\right)=-cos\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}$ tan((π+A))=tan A $tan\left(\mathrm{\pi }+A\right)=tan\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}$ cosec((π+A))=-cosec A$cosec\left(\mathrm{\pi }+A\right)=-cosec\phantom{\rule{0.22em}{0ex}}A$ sec((π+A))=-sec A$\phantom{\rule{0.22em}{0ex}}sec\left(\mathrm{\pi }+A\right)=-sec\phantom{\rule{0.22em}{0ex}}A$ cot((π+A))=cot A$cot\left(\mathrm{\pi }+A\right)=cot\phantom{\rule{0.22em}{0ex}}A$ sin((A+B))=sinAcosB+cosAsinB$sin\left(A+B\right)=sinAcosB+cosAsinB$ sin((A-B))=sinAcosB-cosAsinB$sin\left(A-B\right)=sinAcosB-cosAsinB$ cos((A+B))=cosAcosB-sinAsinB$cos\left(A+B\right)=cosAcosB-sinAsinB$ cos((A-B))=cosAcosB+sinAsinB$cos\left(A-B\right)=cosAcosB+sinAsinB$ tan((A+B))=tanA+tanB1-tanA.tanB$tan\left(A+B\right)=\frac{tanA+tanB}{1-tanA.tanB}$ tan((A-B))=tanA-tanB1+tanA.tanB$tan\left(A-B\right)=\frac{tanA-tanB}{1+tanA.tanB}$ sinA+sinB=2sinA+B2cosA-B2$sinA+sinB=2sin\frac{A+B}{2}cos\frac{A-B}{2}$ sinA-sinB=2cosA+B2sinA-B2$sinA-sinB=2cos\frac{A+B}{2}sin\frac{A-B}{2}$ cosA+cosB=2cosA+B2cosA-B2$cosA+cosB=2cos\frac{A+B}{2}cos\frac{A-B}{2}$ cosA-cosB=-2sinA+B2sinA-B2$cosA-cosB=-2sin\frac{A+B}{2}sin\frac{A-B}{2}$ sin2A=2sinAcosA$sin2A=2sinAcosA$ cos2A=cos2A-sin2A$cos2A=co{s}^{2}A-si{n}^{2}A$ cos2A=2cos2A-1$cos2A=2co{s}^{2}A-1$ cos2A=1-2sin2A$cos2A=1-2si{n}^{2}A$ tan2A=2tanA1-tan2A$tan2A=\frac{2tanA}{1-ta{n}^{2}A}$ sin3A=3sinA-4sin3A$sin3A=3sinA-4si{n}^{3}A$ cos3A=4cos3A-3cosA$cos3A=4co{s}^{3}A-3cosA$ tan3A=3tanA-tan3A1-3tan2A$tan3A=\frac{3tanA-ta{n}^{3}A}{1-3ta{n}^{2}A}$ Principal Solution: the solution of trigonometric equation with unknown angle 'x', where0β©½xβ©½2π$0\beta ©½x\beta ©½2\mathrm{\pi }$ , are called principal solution. General Solution: the solution of trigonometric equation which are generalized by using periodicity are called general solution. if sinπ=0$sin\mathrm{\pi }=0$ , then π=nπ$\phantom{\rule{0.22em}{0ex}}\mathrm{\pi }=n\mathrm{\pi }$ if cosπ=0 $\phantom{\rule{0.22em}{0ex}}cos\mathrm{\pi }=0\phantom{\rule{0.22em}{0ex}}$, then π=((2n+1))π2$\phantom{\rule{0.22em}{0ex}}\mathrm{\pi }=\left(2n+1\right)\frac{\mathrm{\pi }}{2}$ if sinπ=sinπΌ $\phantom{\rule{0.22em}{0ex}}sin\mathrm{\pi }=sin\mathrm{\pi Ό}\phantom{\rule{0.22em}{0ex}}$, then , π=nπ+((-1))nπΌ$\mathrm{\pi }=n\mathrm{\pi }+\left(-1{\right)}^{n}\mathrm{\pi Ό}$ if cosπ=cosπΌ$\phantom{\rule{0.22em}{0ex}}cos\mathrm{\pi }=cos\mathrm{\pi Ό}$ , then , π=2nπΒ±πΌ$\phantom{\rule{0.22em}{0ex}}\mathrm{\pi }=2n\mathrm{\pi }Β±\mathrm{\pi Ό}$ if tanπ=tanπΌ$tan\mathrm{\pi }=tan\mathrm{\pi Ό}$ , then , π=nπ+πΌ$\phantom{\rule{0.22em}{0ex}}\mathrm{\pi }=n\mathrm{\pi }+\mathrm{\pi Ό}$
Trigonometry (Solution of triangle)
 Law of cosinesaa2=b2+c2-2 bc cosAb2=c2+a2-2 ca cosBc2=a2+b2-2 ab cosC$\begin{array}{l}{a}^{2}={b}^{{}^{}2}+{c}^{2}-2\phantom{\rule{0.22em}{0ex}}bc\phantom{\rule{0.22em}{0ex}}cosA\\ {b}^{2}={c}^{{}^{}2}+{a}^{2}-2\phantom{\rule{0.22em}{0ex}}ca\phantom{\rule{0.22em}{0ex}}cosB\\ {c}^{2}={a}^{{}^{}2}+{b}^{2}-2\phantom{\rule{0.22em}{0ex}}ab\phantom{\rule{0.22em}{0ex}}cosC\end{array}$ Law of sinesasinA=bsinB=csinC$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$ Projection Ruleaa=b cosC+c cosBb=c cosA+a cosCc=a cosB+b cosA$\begin{array}{l}a=b\phantom{\rule{0.22em}{0ex}}cosC+c\phantom{\rule{0.22em}{0ex}}cosB\\ b=c\phantom{\rule{0.22em}{0ex}}cosA+a\phantom{\rule{0.22em}{0ex}}cosC\\ c=a\phantom{\rule{0.22em}{0ex}}cosB+b\phantom{\rule{0.22em}{0ex}}cosA\end{array}$ Mollweide's formulaa+bc=cosA-B2sinC2$\frac{a+b}{c}=\frac{cos\frac{A-B}{2}}{sin\frac{C}{2}}$ Napier Analogy (tangent Rule)atanB-C2=b-cb+ccotA2 tanC-A2=c-ac+acotB2 tanA-B2=a-ba+bcotC2$\begin{array}{l}tan\frac{B-C}{2}=\frac{b-c}{b+c}cot\frac{A}{2}\\ \\ tan\frac{C-A}{2}=\frac{c-a}{c+a}cot\frac{B}{2}\\ \\ tan\frac{A-B}{2}=\frac{a-b}{a+b}cot\frac{C}{2}\end{array}$