SOME FORMULA FOR POLYNOMIALS

polynomials are algebraic expressions that consist of variables and coefficients. We can perform arithmetic operations such as addition, subtraction, multiplication and division in a polynomial functions. There are many types of polynomial equations. General form of polynomial equation is: $\\$ $$ a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2}\cdots a_{n-1} x + a_n = 0 $$ It is a $n$ degree polynomial equation, in this equation $(a_0, a_1, a_2,\cdots a_n)$ are the coefficients. for $n$ degree polynomial $a_0 \neq 0$ $\\$ $1.$ If highest degree of $x$ is $1$ , it is called linear equation $\\$ $2.$ If highest degree of $x$ is $2$ , it is called quadratic equation $\\$ $3.$ If highest degree of $x$ is $3$ , it is called cubic equation $\\$ $4.$ If highest degree of $x$ is $4$ , it is called quartic or biquadratic equation $\\$ Let's see some basic formula for solving quadratic and cubic equations $\\$ $$ (a+b)^2 = a^2 + 2ab + b^2 \\ (a-b)^2 = a^2 -2ab+b^2 \\ (a+b)^2 = (a-b)^2 +4ab\\ (a+b)^2 + (a-b)^2 = 2(a^2 +b^2) \\ (a+b)^3 = a^3+b^3+3ab(a+b)\\ (a-b)^3 = a^3-b^3-3ab(a-b)\\ a^2-b^2 = (a+b)(a-b) \\ a^3-b^3 = (a-b)(a^2+b^2+ab) \\ a^3+b^3 = (a+b)(a^2+b^2-ab) \\ (a+b+c)^2 = a^2 +b^2 +c^2+2(ab+bc+ca) \\ (a+b-c)^2 = a^2+b^2+c^2+2(ab-bc-ca) \\ (a-b-c)^2 = a^2+b^2+c^2+2(-ab+bc-ca) \\ (a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a)\\ (a^3+b^3+c^3+3abc) = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) $$ if $(a+b+c)=0$ , then $ a^3+b^3+c^3 = 3abc $.

Integration formula list

 Integral formulas : - Before learning the formulas we have to know that what is integration and why this term is in mathematics field. 

let's know about it - 

The basic means of integration is the process of combining two or more things to make a complete/whole thing. This term used in businesses.

The similar thing is used in mathematics, for calculating a huge term by integrate a little term with a limit. We can compute a complete thing by integrating a little thing using it's limit.

According to the Mathematician Bernhard Riemann,

"Integral is based on a limiting procedure which approximate the area of a curvilinear region by breaking the region into thin vertical slabs."

And integral is also known as inverse form of differentiation.

Now, let us learn formula which is our main motive.

$\int \frac{d}{dx} xdx$ = $x$ $\\$ $\int 1 dx = x + c$, Here $c$ is a constant value. $\\$ $\int x dx = \frac{x^2}{2} + c$$\\$ $\int x^n dx = \frac{x^{n+1}}{n+1} + c$, Here $n \neq -1$ $\\$ $\int \frac{1}{x}dx$ = $log_e x + c$$\\$ integration of trigonometric functions. $\\$ $\int sin x dx = -cos x + c$$\\$ $\int cos x dx = sin x + c$$\\$ $\int sec^2 x dx = tan x + c$$\\$ $\int cosec^2 x = -cot x + c$$\\$ $\int sec x (tan x)dx = sec x + c$ $\\$$\int cosec x (cot x)dx = -cosec x + c $ $\\$$\int tan xdx = log_e |sec x| + c$ $\\$ $\int cot xdx = log_e |sin x| + c$ $\\$ $\int secxdx = log_e |secx + tan x| + c$$\\$ $\int cosec x dx = log_e |cosecx - cot x| + c$$\\$ Integral of some inverse trigonometric functions. $\\$ $\int \frac{1}{\sqrt{1-x^2}}dx = sin^{-1}x + c$$\\$ $\int \frac{1}{1+x^2}dx = tan^{-1}x + c$$\\$ $\int \frac{1}{|x|\sqrt{x^2-1}}dx = sec^{-1}x + c$$\\$ $\int sin^{-1}xdx = x.sin^{-1}x + \sqrt{1-x^2} + c$$\\$ $\int cos^{-1}xdx = x.cos^{-1}x - \sqrt{1-x^2} + c$$\\$ $\int tan^{-1}xdx = x.tan^{-1}x - \frac{1}{2}log_e(1+x^2) + c$$\\$ $\int cot^{-1}xdx = x.cot^{-1}x + \frac{1}{2}log_e(1+x^2) + c$$\\$ $\int sec^{-1}xdx = x.sec^{-1}x - log_e(x+\sqrt{x^2-1}) + c$$\\$ $\int cosec^{-1}xdx = x.cosec^{-1}x + log_e(x+\sqrt{x^2-1}) + c$$\\$ $\\$ Exponential functions :-$\\$ $\int e^{ax}dx = \frac{e^{ax}}{a} + c$$\\$ $\int a^{x}dx = \frac{a^{x}}{log_e(a)} + c$, $a>0,a \neq 1$$\\$ logarithmic functions :-$\\$ $\int log_e(x)dx = x.log_(x)-x + c$ $\\$ Integration for some special functions$\\$ $\int \frac{1}{(x^2-a^2)} dx = \frac{1}{2a}.log_e|\frac{(x-a)}{(x+a)}| + c$$\\$ $\int \frac{1}{(a^2-x^2)} dx = \frac{1}{2a}.log_e|\frac{(a+x)}{(a-x)}| + c$$\\$ $\int \frac{1}{(x^2+a^2)} dx = \frac{1}{a}tan^{-1}(\frac{x}{a}) + c$$\\$ $\int \frac{1}{\sqrt{(x^2-a^2)}} dx = log_e|x + \sqrt{x^2 - a^2}| + c$$\\$ $\int \frac{1}{\sqrt{(a^2-x^2)}} dx = sin^{-1}(\frac{x}{a}) + c$$\\$ $\int \frac{1}{\sqrt{(x^2+a^2)}} dx = log_e|x+\sqrt{x^2 + a^2}| + c$$\\$ $\\$ Hi! learnera $\\$ If you guys find this formula list helpful then please share it as you can.$\\$ Thank you guys$\\$ I will gonna to provide you a technique that can help you to solve huge problems in lesser time as soon as possible. and this may help you to solve NDA/NA question pape, IAF exam, CDS exam, AFCAT and many other exams that contains mathematical questions.

Differentiation formulas

The Differentiation formula list has been provided by me to help all those students who can not take any coaching or tuition, and it also help all readers those want to learn mathematics (Calculus).

This is my first blog that contain differentiation formulas of a constant, trigonometric functions, algebraic  functions, exponential functions, inverse trigonometric functions and logarithmic functions and many more. 

Before reading the formula we have to know "what is differentiation?"

In many colleges and schools students are asked that what is differentiation? Most of the students can not explain so this BLOG is going to be very important for knowing about differentiation.

So let's talk about differentiation:-

Differentiation means rate of change of a function f(x) with respect to its input x . 

and the rate of change is known as derivative of a function with respect to the x.

It can be represented as d/dx.

Now its time to learn formulas of differentiation:- 

$\frac{d}{dx}$ $f(x)$ = $f^{'}{x}$ $\\$ $\\$ $\frac{d}{dx}$ $c = 0$ where $c$ is a constant value. $\\$ $\\$ $\frac{d}{dx}$$(x)$ = $1$ $\\$ $\\$ $\frac{d}{dx}$$x^{n}$ = $n(x)^{n-1}$,$ $ where n is real number.$\\$ $\\$ $\frac{d}{dx} \sin x = \cos x $ $\\$ $\\$ $\frac{d}{dx}$$\cos x$ = $-\sin x$$\\$ $\\$ $\frac{d}{dx}$$\tan x$ = $\sec^{2} x$$\\$ $\\$ $\frac{d}{dx}$$\cot x$ = $-\csc^{2} x$$\\$ $\\$ $\frac{d}{dx}$$\sec x$ = $\sec x \tan x $$\\$ $\\$ $\frac{d}{dx}$$\csc x $ = $-\csc x \cot x $$\\$ $\\$ $\frac{d}{dx}$$\sin^{-1} x$ = $\frac{1}{\sqrt{1-x^{2}}}$$\\$ $\\$ $\frac{d}{dx}$$\cos^{-1} x$ = $-\frac{1}{\sqrt{1-x^{2}}}$$\\$ $\\$ $\frac{d}{dx}$$\tan^{-1} x$ = $\frac{1}{1+x^{2}}$ $\\$ $\\$ $\frac{d}{dx}$$\cot^{-1}x$ = $-\frac{1}{1+x^{2}}$ $\\$ $\\$ $\frac{d}{dx}$$\sec^{-1}x$ = $\frac{1}{|x|\sqrt{1-x^{2}}}$ $\\$ $\\$ $\frac{d}{dx}$$\csc^{-1}x$ = $-\frac{1}{|x|\sqrt{1-x^{2}}}$ $\\$ $\\$ $\frac{d}{dx}$ $e^{x}$ = $e^{x}$ $\\$ $\\$ $\frac{d}{dx}$$a^{x}$ = $a^{x}\ln a $ $\\$ $\\$ $\frac{d}{dx}$$\ln x $ = $\frac{1}{|x|}$ $\\$ $\\$ Derivative of $x$ from the first principle of differentiation. $\\$ $\\$ $\frac{d}{dx}$$f(x)$ = $f^{'}(x)$ = $\lim_{x\to 0}\frac{f(x+h) - f(x)}{h} $ $\\$ $\\$ If there are two functions $u$ and $v$,$\\$ then $\frac{d}{dx}$$(uv)$ = $u\frac{d}{dx}v + v\frac{d}{dx}u$ $\\$ Namaste to my all readers Please Guys share this blog post if this blog post helped you to learn differentiation. wait for my next blog post. My blogs help you to learn calculus from basic to high level. $\\$ THANKS! $\\$ support.