# Antiderivative Formula

Anything that is an opposite of a function and has been differentiated in trigonometric terms is known as an anti-derivative. Both the antiderivative and the differentiated function are continuous on a specified interval. In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. Some of the formulas are mentioned below.

## Basic Antiderivatives

$\large If\: f(x) = a,\: then \: F(x) = ax+C$
$\large If\: f(x) = x^{a},\: then \: F(x) = \frac{x^{a+1}}{a+1} + C\:(unless\:a=-1)$
$\large If\: f(x) = \frac{1}{x},\: then \: F(x) = \ln (x)+C$
$\large If\: f(x) = e^{x},\:then\:F(x) = e^{x}+C$
$\large If\: f(x) = \cos(x),\:then\:F(x) = \sin(x)+C$
$\large If\: f(x) = \sin(x),\:then\:F(x) = -\cos(x)+C$
$\large If\: f(x) = \sec^{2}(x),\:then\:F(x) = \tan(x)+C$

$\large \int e^{x}dx=e^{x}+C$
$\large \int a^{x}dx=\frac{a^{x}}{\ln a}+C$
$\large \int \frac{1}{x}dx=\ln\left | x \right |+C$
$\large \int \cos x\:dx= \sin x +C$
$\large \int \sec ^{2}x\:dx= \tan x +C$
$\large \int \sin x\:dx= -\cos x +C$
$\large \int \csc^{2} x\:dx= -\cot x +C$
$\large \int \sec x\:\tan x\:dx= \sec x +C$
$\large \int \frac{1}{1+x^{2}}\:dx= \arctan x +C$
$\large \int \frac{1}{\sqrt{1+x^{2}}}\:dx= \arcsin x +C$
$\large \int \csc x \cot x\:dx= -\csc x +C$
$\large \int\sec x\:dx= \ln \left | \sec x+\tan x \right | +C$
$\large \int\csc x\:dx= -\ln \left |\csc x+\cot x \right | +C$
$\large \int x^{n}\:dx= \frac{x^{n+1}}{n+1}+C,\: when\:n\neq -1$
$\large \int \sinh x \:dx= \cosh x+C$
$\large \int \cosh x \:dx= \sinh x+C$

## Antiderivative Rules

$\large If\:the\:antiderivative\:of\:f(x)\:is\:F(x), and\:the\:antiderivative\:of\:g(x)\:G(x),\:then\:$
1) $\large The\:Antiderivative\:of\:af(x)+bg(x)\:is\:aF(x)+bG(x)\:(for\:any\:a,b)$
2) $\large The\:Antiderivative\:of\:f(ax+b)\:is\:\frac{1}{a}F(ax+b)$

$Therefore,$

$\large If\:f(x)=(dx+b)^{a}\:then\:F(x)=\frac{1}{d}\frac{(dx+b)^{a+1}}{a+1} +C\:(unless\:a=-1)$
$\large If\:f(x)=\frac{1}{ax+b}\:then\:F(x)=\frac{1}{a}\ln (ax+b)+C$
$\large If\:f(x)=e^{ax+b}\:then\:F(x)=\frac{1}{a}e^{ax+b}+C$
$\large If\:f(x)=\cos(ax+b)\:then\:F(x)=\frac{1}{a}\sin(ax+b)+C$
$\large If\:f(x)=\sin(ax+b)\:then\:F(x)=-\frac{1}{a}\cos(ax+b)+C$
$\large If\:f(x)=\sec^{2}(ax+b)\:then\:F(x)=\frac{1}{a}\tan(ax+b)+C$

#### Practise This Question

A car emitting sound of frequency 500 Hz speeds towards a fixed wall at 4 m/s. An observer in the car hears both the source frequency as well as the frequency of sound reflected from the wall. If he hears 10 beats per second between the two sounds, the velocity of sound in air will be