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 $\displaystyle \frac{\pi }{6} \text{ to } \frac{\pi}{3}$


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