Question

What is the integral of a constant?

Answer 1

In this type of question we have to use the concepts of integration and some rules of indices. We know that anything raised to zero is always equal to 1 that is ${{x}^{0}}=1$. Also we know that $\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1}+c}$ where $c$ is an arbitrary constant which is also known as constant of integration

Complete step-by-step answer:
Now we have to find the integral of a constant say k.
For this let us consider,
$= \int{kdx}$
Now as k is a constant we can rewrite the integral as
$= k\int{1dx}$
By the rule of indices we can write ${{x}^{0}}=1$ and hence
$= k\int{{{x}^{0}}dx}$
By using the rule, $\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1}+c}$ we get,
$= k\left( \dfrac{{{x}^{0+1}}}{0+1} \right)+c$ where $c$ is an arbitrary constant which is also known as constant of integration
$= k\left( \dfrac{{{x}^{1}}}{1} \right)+c$
On simplifying we can write,
$= kx+c$
Hence, we can say that the integral of a constant say k is given by, $kx+c$
In other words, we can write, $\int{kdx=kx+c}$

Note: In this type of question one of the students may use another way to find the integral of a constant in following manner:
We know that by the rules of differentiation $\dfrac{d}{dx}\left( kx+c \right)=k\dfrac{d}{dx}\left( x \right)+\dfrac{d}{dx}\left( c \right)=k$ where $k$ and $c$ are constant and we know that the derivative of a constant is zero.
$= \dfrac{d}{dx}\left( kx+c \right)=k$
Now taking integral of both sides we get,
$= \int{\left[ \dfrac{d}{dx}\left( kx+c \right) \right]dx}=\int{kdx}$
As we know that integration and differentiation are inverse of each other,
$= kx+c=\int{kdx}$
Hence, we can say that the integral of a constant say k that is $\int{kdx}$ equals to $kx+c$ where $c$ is an arbitrary constant which is also known as constant of integration.