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.
