Question

What is the integral of a constant?

Answer 1

Hint: 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 $$\int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}+c$$ 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, $$\int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}+c$$ we get,
$$=k\left({\displaystyle \frac{x^{0+1}}{0+1}}\right)+c$$ where $$c$$ is an arbitrary constant which is also known as constant of integration
$$=k\left({\displaystyle \frac{x^1}{1}}\right)+c$$
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 $${\displaystyle \frac{d}{dx}}(kx+c)=k{\displaystyle \frac{d}{dx}}\left(x\right)+{\displaystyle \frac{d}{dx}}\left(c\right)=k$$ where $$k$$ and $$c$$ are constant and we know that the derivative of a constant is zero.
$$={\displaystyle \frac{d}{dx}}(kx+c)=k$$
Now taking integral of both sides we get,
$$=\int \left[{\displaystyle \frac{d}{dx}}(kx+c)\right]dx=\int kdx$$
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.