Question

What is the integral of $3x?$

Answer 1

We know that the integral of a number is the antiderivative of that number. When the integration has upper limit and lower limit, we call the integration definite integration and when the integration does not have upper limit and lower limit, we call the integration indefinite integration. We should remember that, in the indefinite integration, we should add the constant of integration after the integration is done.

Complete step-by-step answer:
Let us consider the given function $3x.$
We are asked to integrate the given function.
Since the integration has no upper limit and lower limit, this integration is indefinite integration.
We know the basic integration rules. So, when we integrate a function that is a product of a constant and a variable with respect to the variable, the constant can be taken out of the integral symbol and then the variable alone is treated under the integral sign.
We know that $\int{{{x}^{n}}}dx=\dfrac{{{x}^{n+1}}}{n+1}+C$ where $C$ is the constant of integration.
We also know that $\int{af\left( x \right)}dx=a\int{f\left( x \right)}dx+C.$
So, from the above property, we will get $\int{3x}dx=3\int{x}dx+C$
Using the above identity, we will get $\int{3x}dx=3\int{x}dx+C=3\dfrac{{{x}^{1+1}}}{1+1}+C=\dfrac{3}{2}{{x}^{2}}+C.$
Hence the integral is $\int{3x}dx=\dfrac{3}{2}{{x}^{2}}+C.$

Note: We know that we can find the area of a circular region, rectangular region and such regions that are uniformly plotted can be found by certain formulas. But if we are given a non-uniform region and asked to find the area, we can use integration to get the result.