Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 42

Determine the integral $\displaystyle \int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx$

Using Integration by parts

$\int \csc^3 x dx = \int udv$

where


$
\begin{equation}
\begin{aligned}

dv =& \csc^2 x dx
\\
\\
v =& - \cot x
\\
\\
u =& \csc x
\\
\\
du =& - \csc x \cot x dx

\end{aligned}
\end{equation}
$


then


$
\begin{equation}
\begin{aligned}

\int \csc^3 x dx =& \int u dv
\\
\\
\int \csc^3 x dx =& uv - \int vdu
\\
\\
\int \csc^3 x dx =& - \csc x \cot x - \int - \cot x \cdot - \csc x \cot x dx
\\
\\
\int \csc^3 x dx =& - \csc x \cot x - \int \csc x \cot^2 x dx
\qquad \text{Apply Trigonometric Identity } \csc^2 x = 1 + \cot^2 x \text{ for } \cot^2 x
\\
\\
\int \csc^3 x dx =& - \csc x \cot x - \int \csc x (\csc^2 x - 1) dx
\\
\\
\int \csc^3 x dx =& - \csc x \cot x - \left( \int (\csc^3 x) - \csc x) dx \right)
\\
\\
\int \csc^3 x dx =& - \csc x \cot x - \int \csc^3 x dx + \int \csc x dx
\qquad \text{Combine like terms}


\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\int \csc^3 x dx + \int \csc^3 x dx =& - \csc x \cot x + \int \csc x dx
\\
\\
2 \int \csc^3 x dx =& - \csc x \cot x + \int \csc x dx
\\
\\
\int \csc^3 x dx =& \frac{- \csc x \cot x + \int \csc x dx}{2}
\\
\\
\int \csc^3 x dx =& \frac{-1}{2} \csc x \cot x + \frac{1}{2} \int \csc x dx
\\
\\
\int \csc^3 x dx =& \frac{-1}{2} \csc x \cot x + \frac{1}{2} (- \ln (\csc x + \cot x)) + c
\\
\\
\int \csc^3 x dx =& \frac{-1}{2} \csc x \cot x - \frac{1}{2} \ln (\csc x + \cot x) + c


\end{aligned}
\end{equation}
$


Evaluating the limit from $\displaystyle \frac{\pi }{6} \text{ to } \frac{\pi}{3}$


$
\begin{equation}
\begin{aligned}

\int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx =& \left[ \frac{-1}{2} \csc x \cot x - \frac{1}{2} \ln (\csc x + \cot x) \right]^{\frac{\pi}{3}}_{\frac{\pi}{6}}
\\
\\
\int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx =& \frac{-1}{2} \csc \left( \frac{\pi}{3} \right) \cot \left( \frac{\pi}{3} \right) - \frac{1}{2} \ln \left( \csc \left( \frac{\pi}{3} \right) + \cot \left( \frac{\pi}{3} \right) \right) + \frac{1}{2} \csc \left( \frac{\pi}{6} \right) \cot \left( \frac{\pi}{6} \right) + \frac{1}{2} \ln \left( \csc \left( \frac{\pi}{6} \right) + \cot \left( \frac{\pi}{6} \right) \right)
\\
\\
\int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx =& \frac{-1}{\cancel{2}} \left( \frac{\cancel{2} \sqrt{3}}{3} \right) \left( \frac{\sqrt{3}}{3} \right) - \frac{1}{2} \ln \left( \frac{2 \sqrt{3}}{3} + \frac{\sqrt{3}}{3} \right) + \frac{1}{\cancel{2}} (\cancel{2}) (\sqrt{3}) + \frac{1}{2} \ln (2 + \sqrt{3})
\\
\\
\int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx =& \frac{-3}{9} - \frac{1}{2} \ln \left( \frac{\cancel{3} \sqrt{3}}{\cancel{3}} \right) + \sqrt{3} + \frac{1}{2} \ln (2 + \sqrt{3})
\\
\\
\int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \csc^3 x dx =& \frac{-1}{3} - \frac{1}{2} \ln (\sqrt{3}) + \sqrt{3} + \frac{1}{2} \ln (2 + \sqrt{3})

\end{aligned}
\end{equation}
$