Math Doubts

Proof of Integral of csc²x formula

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.

Differentiation of Cot function

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

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

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

Insert an Arbitrary constant

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.

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

Integral of csc²x function

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.

Math Doubts

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

Maths Topics

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

Maths Problems

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

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved