Antiderivative Formula

Anything that is the 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

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

These can be written using integral as given below:

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

Antiderivative Rules

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


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

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