Integral Rule of Cosecant function

Formula

$\displaystyle \int{\csc{x}}\,dx$ $\,=\,$ $\log_{e}{|\csc{x}-\cot{x}|}+c$

The integral of cosecant is equal to the natural logarithm of the subtraction of cotangent from cosecant.

Introduction

The cosecant function is written as either $\csc{x}$ or $\operatorname{cosec}{x}$ in trigonometric mathematics when a variable $x$ represents angle of a right triangle.

The indefinite integral of cosecant of angle $x$ with respect to $x$ is written in the following forms in mathematics.

$(1).\,\,\,$ $\displaystyle \int{\csc{x}}\,dx$

$(2).\,\,\,$ $\displaystyle \int{\operatorname{cosec}{x}}\,dx$

The integral of the cosecant of angle $x$ with respect to $x$ is equal to the natural logarithm of the subtraction of cot of angle $x$ from cosecant of angle $x$, and plus the constant of integration.

$\implies$ $\displaystyle \int{\csc{x}}\,dx$ $\,=\,$ $\log_{e}{|\csc{x}-\cot{x}|}+c$

According to the logarithms, the natural logarithm is also written in the below form simply in mathematics.

$\implies$ $\displaystyle \int{\csc{x}}\,dx$ $\,=\,$ $\ln{|\csc{x}-\cot{x}|}+c$

Alternative forms

The integral rule of cosecant function can also be written in terms of any variable in mathematics.

$(1).\,\,\,$ $\displaystyle \int{\csc{u}}\,du$ $\,=\,$ $\log_{e}{|\csc{u}-\cot{u}|}+c$

$(2).\,\,\,$ $\displaystyle \int{\csc{t}}\,dt$ $\,=\,$ $\log_{e}{|\csc{t}-\cot{t}|}+c$

$(3).\,\,\,$ $\displaystyle \int{\csc{y}}\,dy$ $\,=\,$ $\log_{e}{|\csc{y}-\cot{y}|}+c$

Proof

Learn how to derive the integration formula for cosecant function in integral calculus.