Math Doubts

Integral of csc²x formula

Formula

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

Introduction

Consider, $x$ is a variable, used to represent an angle of a right triangle. Then, the square of the cosecant of angle $x$ is written as $\csc^2{x}$ or $\operatorname{cosec}^2{x}$. Now, the indefinite integral of cosecant squared function with respect to $x$ is written in integral calculus as the following mathematical form.

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

The indefinite integral of cosecant squared of angle $x$ function with respect to $x$ is equal to sum of the negative cotangent of angle $x$ and a constant of integration.

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

Alternative forms

The indefinite integration of cosecant squared function formula can be written in terms of any variable.

$(1) \,\,\,$ $\displaystyle \int{\csc^2{(p)} \,} dp \,=\, -\cot{(p)}+c$

$(2) \,\,\,$ $\displaystyle \int{\csc^2{(y)} \,} dy \,=\, -\cot{(y)}+c$

$(3) \,\,\,$ $\displaystyle \int{\csc^2{(k)} \,} dk \,=\, -\cot{(k)}+c$

Proof

Learn how to derive the integration of cosecant squared function rule in integral calculus.



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