Integration of Cosecant Squared X

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

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

Other Integral Formulas of Cosecant Tangent Function:

The other formulas of cosecant squared of x integral with angle is in the form of 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