# Integration of Cosecant Squared X

Integration of cosecant squared of x is an important integral formula in integral calculus, and this integral belongs to the trigonometric formulae.

The integration of cosecant squared of x is of the form

$\int {{{\csc }^2}xdx = } – \cot x + c$

To prove this formula, consider

$\frac{d}{{dx}}\left[ { – \cot x + c} \right] = – \frac{d}{{dx}}\cot x + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}\cot x = – {\csc ^2}x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ { – \cot x + c} \right] = – \left( { – {{\csc }^2}x} \right) + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ { – \cot x + c} \right] = {\csc ^2}x \\ \Rightarrow {\csc ^2}x = \frac{d}{{dx}}\left[ { – \cot x + c} \right] \\ \Rightarrow {\csc ^2}xdx = d\left[ { – \cot x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{{\csc }^2}xdx} = \int {d\left[ { – \cot x + c} \right]}$

As we know that by definition integration is the inverse process of the derivative, the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.
$\int {{{\csc }^2}xdx = } – \cot x + c$

Other Integral Formulae of the Cosecant Tangent Function

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

1. $\int {{{\csc }^2}axdx = – \frac{{\cot ax}}{a}} + c$

2. $\int {{{\csc }^2}f\left( x \right)f’\left( x \right)dx = – \cot f\left( x \right) + c}$

Example: Evaluate the integral $\int {{{\csc }^2}8xdx}$ with respect to $x$

We have integral $I = \int {{{\csc }^2}8xdx}$

Using the formula $\int {{{\csc }^2}axdx = – \frac{{\cot ax}}{a}} + c$, we have
$I = \int {{{\csc }^2}8x} = – \frac{{\cot 8x}}{8} + c$