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

Integration

Indefinite Integrals

Primitive Or Anti derivative

Indefinite Integrals

Limits

Continuity

Differentiability

Differentiation

Tangents and Normals

Increasing and Decreasing Functions

Maxima and Minima

Indefinite Integrals

Definite Integrals

Improper integral

Differential Equations

Learn to Think