Assume, $x$ as a variable and represents angle of a right triangle. In trigonometry, the cosecant squared of angle $x$ is written as $\csc^2{x}$ or $\operatorname{cosec}^2{x}$ in mathematical form. The indefinite integration of cosecant squared function with respect to $x$ is written as the following mathematical form in differential calculus

$\displaystyle \int{\csc^2{x} \,}dx \,\,\,$ (or) $\,\,\, \displaystyle \int{\operatorname{cosec}^2{x} \,}dx$

Now, let us start deriving the integration formula for the cosecant squared function in integral calculus.

Express the derivative of cot function formula with respect to $x$ in mathematical form.

$\dfrac{d}{dx}{\, \cot{x}} \,=\, -\csc^2{x}$

$\implies$ $\dfrac{d}{dx}{\, (-\cot{x})} \,=\, \csc^2{x}$

As per differential calculus, the derivative of a constant is always zero. So, there is no problem in adding an arbitrary constant to cot function.

$\implies$ $\dfrac{d}{dx}{(-\cot{x}+c)} \,=\, \csc^2{x}$

According to integral calculus, the collection of all primitives of $\csc^2{x}$ function is called the integration of $\csc^2{x}$ function. It can be written in mathematical form in two ways.

$\displaystyle \int{\csc^2{x} \,}dx \,\,\,$ (or) $\,\,\, \displaystyle \int{\operatorname{cosec}^2{x} \,}dx$

The primitive or an antiderivative of $\csc^2{x}$ function is $-\cot{x}$ and the constant of integration ($c$) in this case.

$\dfrac{d}{dx}{(-\cot{x}+c)} = \csc^2{x}$ $\,\Longleftrightarrow\,$ $\displaystyle \int{\csc^2{x} \,}dx = -\cot{x}+c$

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\csc^2{x} \,}dx = -\cot{x}+c$

Therefore, it has proved that the indefinite integration of cosecant squared of an angle function is equal to the sum of the negative cot function and a constant of integration.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved