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

There are 3 cab companies A, B and C. They provide cab service at different time slots. The time taken for the trip depends upon the driver skill. The bar graph shown below gives the details.

The time taken for cab C1 of company A to reach the class is ______.