# Proof of Reciprocal Rule of Integration

Let $x$ be a variable, which represents a non-zero real number. The multiplicative inverse or reciprocal of variable is written as $\dfrac{1}{x}$ mathematically. The indefinite integral of $\dfrac{1}{x}$ with respect to $x$ is written in integral calculus in the following mathematical form.

$\displaystyle \int{\dfrac{1}{x} \,}dx$

### Derivative of logarithmic function

According to differentiation, the derivative of natural logarithm of $x$ with respect to $x$ is equal to multiplicative inverse or reciprocal of $x$. So, it is written in the following mathematical form.

$\dfrac{d}{dx}{\, \log_e{x}} \,=\, \dfrac{1}{x}$

$\implies$ $\dfrac{d}{dx}{\ln{x}} \,=\, \dfrac{1}{x}$

### Inclusion of an Arbitrary constant

The derivative of a constant is zero as per differential calculus. So, it does not affect the differentiation when an arbitrary constant is included in differentiation by adding it to natural logarithmic function.

$\implies$ $\dfrac{d}{dx}{(\ln{x}+c)} \,=\, \dfrac{1}{x}$

### Integral of logarithmic function

According to integral calculus, the collection of all primitives of multiplicative inverse function $\dfrac{1}{x}$ is called the indefinite integral of $\dfrac{1}{x}$ function with respect to $x$ and it is written in the following mathematical form in mathematics.

$\displaystyle \int{\dfrac{1}{x} \,}dx$

Actually, the primitive or an antiderivative of $\dfrac{1}{x}$ is $\log_e{x}$ and the arbitrary constant becomes the constant of integration.

$\dfrac{d}{dx}{(\ln{x}+c)} = \dfrac{1}{x}$ $\,\Longleftrightarrow\,$ $\displaystyle \int{\dfrac{1}{x} \,}dx = \log_e{x}+c$

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\dfrac{1}{x} \,}dx = \ln{x}+c$

Therefore, it has proved that the anti-derivative or indefinite integration of multiplicative inverse or reciprocal function is equal to the sum of the natural logarithm and the constant of integration.

Latest Math Topics
Jun 26, 2023
Jun 23, 2023

Latest Math Problems
Jul 01, 2023
Jun 25, 2023
###### Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Practice now

###### Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

###### Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.