Indefinite Integrals

Primitive Or Anti derivative

A function \displaystyle \phi \left( x \right)  is called a primitive of a function f(x) if \displaystyle \phi `\left( x \right) =f\left( x \right)

For example, \displaystyle \frac { { x }^{ 4 } }{ 4 }  is primitive of \displaystyle { x }^{ 3 }  beacause \displaystyle \frac { d }{ dx } \left( \frac { { x }^{ 4 } }{ 4 } \right) ={ x }^{ 3 }

Indefinite Integrals

If f(x) is any anti-derivative or primitive of a function f(x) then the most general anti-derivative of f(x) is called an indefinite integral.

Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by \displaystyle \int { f\left( x \right) } dx    Thus,

\displaystyle \frac { d }{ dx } \left( \phi \left( x \right) +C \right) =f\left( x \right) \Leftrightarrow \int { f\left( x \right) } dx=\phi \left( x \right) dx\quad +C\quad ......\left( i \right) 

where \displaystyle \phi \left( x \right)  is primitive of f(x) and C is an arbitrary constant known as the constant of integration.

Here \displaystyle \int   is integral sign , f(x) is the integrated , x is the variable of integration and d x is the element of integration of differential of x.

The process of finding an indefinite integral of a given function is called integration of the function.

It follows from the above discussion that integrating a function f(x) means finding a function \displaystyle \phi \left( x \right)  such that \displaystyle \frac { d }{ dx } \left( \phi \left( x \right) \right) =f\left( x \right)





April 18, 2019
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