Cosecant Integral Formula

In this tutorial we will prove the formula of cosecant integral which is another important formula in integral calculus. This integral belongs to the category of trigonometric integral formulae.

The integration of the cosecant function is of the form
\[\int {\csc xdx = } \ln \left( {\csc x – \cot x} \right) + c = \ln \tan \left( {\frac{x}{2}} \right) + c\]

To prove this formula, consider
\[\int {\csc x} dx = \int {\frac{{\csc x\left( {\csc x – \cot x} \right)}}{{\csc x – \cot x}}} dx\]

By multiplying and dividing the relation $$\left( {\csc x – \cot x} \right)$$
\[\begin{gathered} \int {\csc x} dx = \int {\frac{{{{\csc }^2}x – \csc x\cot x}}{{\csc x – \cot x}}} dx \\ \Rightarrow \int {\csc x} dx = \int {\frac{{ – \csc x\cot x + {{\csc }^2}x}}{{\csc x – \cot x}}} dx \\ \end{gathered} \]

Here $$f\left( x \right) = \csc x – \cot x$$, then $$f’\left( x \right) = – \csc x\cot x + {\csc ^2}x$$

Now using the formula of integration $$\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}dx = \ln f\left( x \right) + c} $$, we have
\[\int {\csc x} dx = \ln \left( {\csc x – \cot x} \right) + c\]

Now we can further solve this result as follows:
\[\begin{gathered} \int {\csc x} dx = \ln \left( {\frac{1}{{\sin x}} – \frac{{\cos x}}{{\sin x}}} \right) + c = \ln \left[ {\frac{{1 – \cos x}}{{\sin x}}} \right] + c \\ \Rightarrow \int {\csc x} dx = \ln \left[ {\frac{{2{{\sin }^2}\left( {\frac{x}{2}} \right)}}{{2\sin \left( {\frac{x}{2}} \right)\cos \left( {\frac{x}{2}} \right)}}} \right] + c \\ \Rightarrow \int {\csc x} dx = \ln \left[ {\frac{{\sin \left( {\frac{x}{2}} \right)}}{{\cos \left( {\frac{x}{2}} \right)}}} \right] + c \\ \Rightarrow \int {\csc x} dx = \ln \tan \left( {\frac{x}{2}} \right) + c \\ \end{gathered} \]

In conclusion, we can write this formula as
\[\int {\csc xdx = } \ln \left( {\csc x – \cot x} \right) + c = \ln \tan \left( {\frac{x}{2}} \right) + c\]

Other Integral Formulae of the Cosecant Function

The other formulae of cosecant integral with an angle of sine in the form of a function are given as

1. \[\int {\csc axdx = } \frac{1}{a}\ln \left( {\csc ax – \cot ax} \right) + c = \frac{1}{a}\ln \tan \left( {\frac{{ax}}{2}} \right) + c\]

2. \[\int {\csc f\left( x \right)f’\left( x \right)dx = \ln \left[ {\csc f\left( x \right) – \cot f\left( x \right)} \right] + c} = \ln \tan \left[ {\frac{{f\left( x \right)}}{2}} \right] + c\]