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.