Ucale

# 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
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.   