Integration of Cscx Cotx

Integration of the cosecant cotangent function is an important integral formula in integral calculus, and this integral belongs to the 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 the derivative, so the integral sign \int {} and \frac{d}{{dx}} on the right side will cancel each other out, i.e.

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

Other Integral Formulae of the Cosecant Cotangent Function

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

1.

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

2.

\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