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

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