Antiderivative

Summary

Given: a function \(f\)
An antiderivative of \(f(x)\) is a new function, typically denoted by \(F\left( x \right)\) , satisfying the condition \(F'\left( x \right)=f\left( x \right)\) .
Another common name for the antiderivative \(F\) : the primitive function of \(f\) .
Example: \(F\left( x \right)=x^2\) is an antiderivative or primitive function of \(f\left( x \right)=2x\) since \(F'\left( x \right)=f\left( x \right)\) .
If \(F\left( x \right)\) is an antiderivative of \(f\left( x \right)\) then so is \(F\left( x \right)+C\) for any constant \(C\) and \(f\) can have no other antiderivatives.
The constant “ \(C\) ” is called the constant of integration .
\(F\left( x \right)+C\) is typically called the indefinite integral of \(f\) , denoted by

\[\int{ f\left( x \right)dx }=F\left( x \right)+C\]

\(∫\) is called the integral symbol, \(f\left( x \right)\) is called the integrand , \(x\) is called the variable of integration .
Example

\[\int{ 2xdx }=x^2+C\]

\[ \frac{d}{dx}\int{ f\left( x \right)dx }= \frac{d}{dx}\left( F\left( x \right)+C \right)=f\left( x \right)\]

\[\int{ \frac{d}{dx}f\left( x \right)dx }=f\left( x \right)+C\]

Fundamental theorem of calculus, second form. If \(f:[a,b]→R\) is a continuous function and \(F\) is a primitive function of \(f\) then

\[\int_{a}^{b}{ ‍ }f(x)dx=F(b)-F(a)\]

Fundamental theorem of calculus, first form. If \(f\) is continuous on \(I=\left[ a,b \right]\) and \(a≤x≤b\) then the definite integral

\[F(x)=\int_{a}^{x}{ ‍ }f(u)du\]

\[ \frac{d}{dx}F\left( x \right)= \frac{d}{dx}\int_{a}^{x}{ ‍ }f\left( u \right)du=f\left( x \right)\]