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

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