Integration of Cscx Cotx

Integration of cosecant cotangent function is an important integral formula in integral calculus; this integral belongs to trigonometric formulae.

The integration of cosecant cotangent is of the form

\int  {\csc x\cot xdx = }  - \csc x + c

To prove this formula, consider

\frac{d}{{dx}}\left[  {\csc x + c} \right] = \frac{d}{{dx}}\csc x + \frac{d}{{dx}}c

Using the derivative formula \frac{d}{{dx}}\csc x =  - \csc x\cot x, we have

\begin{gathered} \frac{d}{{dx}}\left[ {\csc x + c} \right]  =  - \csc x\cot x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\csc x +  c} \right] =  - \csc x\cot x \\ \Rightarrow   - \csc x\cot x = \frac{d}{{dx}}\left[ {\csc x + c} \right] \\ \Rightarrow \csc x\cot x =  - \frac{d}{{dx}}\left[ {\csc x + c} \right] \\ \Rightarrow \csc x\cot xdx =  - d\left[ {\csc x + c} \right]\,\,\,\,{\text{  -  -   - }}\left( {\text{i}} \right) \\ \end{gathered}

Integrating both sides of equation (i) with respect to x, we have

\int  {\csc x\cot xdx}  =  - \int {d\left[ {\csc x + c} \right]}

As we know that by definition integration is the inverse process of derivative, so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, i.e.

\int  {\csc x\cot xdx = }  - \csc x + c

Other Integral Formulas of Cosecant Cotangent Function:
The other formulas of cosecant tangent integral with angle is in the form of function are given as


\int  {\csc ax\cot axdx =  - \frac{{\csc ax}}{a}}  + c


\int  {\csc f\left( x \right)\cot f\left( x \right)f'\left( x \right)dx =  - \csc f\left( x \right) + c}

Example: Evaluate the integral \int {\csc 9x\cot 9xdx}  with respect to x

We have integral

I =  \int {\csc 9x\cot 9xdx}

Using the formula \int  {\csc ax\cot axdx =  - \frac{{\csc  ax}}{a}}  + c, we have

I =  \int {\csc 9x\cot 9xdx}  =  - \frac{{\csc 9x}}{9} + c