Question

How do you integrate \[\dfrac{2}{x}\]?

Answer 1

Hint: Here we will use the basic concept of the integration. First, we will take the constant out of the integration. Then we will integrate the variable part using a suitable integration formula to get the required answer.

Complete step by step solution:
Given equation is \[\dfrac{2}{x}\].
Let \[I\] be the integration of the given equation. Therefore we can write the integration of the equation as
\[I = \int {\dfrac{2}{x}dx} \]
Now we will take the constant terms out of an integral part of the equation. So, we will take 2 outside of an integral part of the equation. Therefore, we get
\[ \Rightarrow I = 2\int {\dfrac{1}{x}dx} \]
We know that the integration of the \[\dfrac{1}{x}\] is equal to \[\ln \left| x \right|\] i.e. \[\int {\dfrac{1}{x}dx} = \ln \left| x \right|\]. Therefore by putting the value of this integration in the above equation, we get
\[ \Rightarrow I = 2\ln \left| x \right| + C\]
So we can write the above integration as
\[ \Rightarrow I = \int {\dfrac{2}{x}dx} = 2\ln \left| x \right| + C\]

Hence, the integration of \[\dfrac{2}{x}\] is equal to \[2\ln \left| x \right| + C\] where C is the constant.

Note:
Integration is defined as the summation of all the discrete data. Differentiation is the opposite of integration i.e. differentiation of the integration is equal to the value of the function or vice versa. Integration is used to find the area inside a curve of a function or we can say that integration is used to find the function which denotes displacement, area, volume, etc. We must not forget to put the constant term \[C\] after the integration of an equation. The value of the constant can be anything i.e. it can be zero or any value. We should remember the basic integration of the common functions and forms of equations to solve the question easily.