Cosecant Integral Formula

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

The integration of 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 further we can 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}

So 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 Formulas of Cosecant Function:
The other formulas of cosecant integral with angle of sine is in the form of function are given as


\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


\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