Integral of Cosecant Cubed X

In this tutorial we shall discuss the integral of cosecant cubed of $$X$$, and this integration can be evaluated by using integration by parts. We first break the power of the function from cube into square and one power of the function. Then we can use integration by parts of that function because we know the formula for integration cosecant square of $$X$$ from previous tutorials.

The integral of the cosecant cubed of $$X$$ is of the form
\[I = \int {{{\csc }^3}xdx} \,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

First we break $${\csc ^3}x$$ into $$\csc x$$ and $${\csc ^2}x$$, and now the integral (i) becomes
\[I = \int {\csc x{{\csc }^2}xdx} \,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]

$$\csc x$$ and $${\csc ^2}x$$ are the first and second functions that form integration by parts. Using the formula for integration by parts, we have
\[\int {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int {g\left( x \right)dx – \int {\left[ {\frac{d}{{dx}}f\left( x \right)\int {g\left( x \right)dx} } \right]dx} } } \]

Using the formula above, equation (ii) becomes:
\[\begin{gathered} I = \csc x\int {{{\csc }^2}xdx – \int {\left[ {\frac{d}{{dx}}\csc x\left( {\int {{{\csc }^2}xdx} } \right)} \right]} dx} \\ \Rightarrow I = – \csc x\cot x – \int {\left[ {\left( { – \csc x\cot x} \right)\left( { – \cot x} \right)} \right]} dx \\ \Rightarrow I = – \csc x\cot x – \int {\left[ {\csc x{{\cot }^2}x} \right]} dx \\ \end{gathered} \]

Now using trigonometric identity $${\cot ^2}x = {\csc ^2}x – 1$$, we have
\[\begin{gathered} I = – \csc x\cot x – \int {\left[ {\csc x\left( {{{\csc }^2}x – 1} \right)} \right]} dx \\ \Rightarrow I = – \csc x\cot x – \int {\left[ {{{\csc }^3}x – \csc x} \right]} dx \\ \Rightarrow I = – \csc x\cot x – \int {{{\csc }^3}xdx + \int {\csc xdx} } \\ \end{gathered} \]

From our original problem $$I = \int {{{\csc }^3}xdx} $$, using this value, we have
\[\begin{gathered} I = – \csc x\cot x – I + \ln \left( {\csc x – \cot x} \right) + c \\ \Rightarrow I + I = – \csc x\cot x + \ln \left( {\csc x – \cot x} \right) + c \\ \Rightarrow I = \frac{1}{2}\left[ { – \csc x\cot x + \ln \left( {\csc x – \cot x} \right)} \right] + c \\ \Rightarrow \int {{{\csc }^3}xdx} = \frac{1}{2}\left[ { – \csc x\cot x + \ln \left( {\csc x – \cot x} \right)} \right] + c \\ \end{gathered} \]