Question

What is the integral of $\dfrac{1}{2x}$ ?

Answer 1

Hint : We first explain the term $\dfrac{dy}{dx}$ where $y=f\left( x \right)$ . We then need to integrate the equation$\int{\dfrac{1}{2x}dx}$ once to find all the solutions of the differential equation. We take one constant for the integration. We get the equation of a logarithmic function.

Complete step by step solution:
We have to find the integral of the equation $\dfrac{1}{2x}$ . The mathematical form is $\int{\dfrac{1}{2x}dx}$.
The main function is $y=f\left( x \right)$ .
We have to find the antiderivative or the integral form of the equation.
We know the integral form of $\int{\dfrac{1}{x}dx}=\log \left| x \right|+c$.
Constant terms get separated from the integral.
Simplifying the differential form,
We get $\int{\dfrac{1}{2x}dx}=\dfrac{1}{2}\int{\dfrac{1}{x}dx}=\dfrac{1}{2}\log \left| x \right|+c$.
Here $c$ is another constant.
The integral form of the equation $\dfrac{1}{2x}$ is $\dfrac{1}{2}\log \left| x \right|+c$.
So, the correct answer is “$\dfrac{1}{2}\log \left| x \right|+c$”.

Note : The solution of the differential equation is the equation of a logarithmic function. The first order differentiation of $\dfrac{1}{2}\log \left| x \right|+c$ gives the tangent of the circle for a certain point which is equal to $\dfrac{dy}{dx}=\dfrac{1}{2x}$ .