Chapter 7 - Section 7.2 - Trigonometric Integrals - 7.2 Exercises - Page 485: 40

$$\sqrt 3 - \frac{1}{3} + \frac{1}{2}\ln \left( {\frac{{3 + 2\sqrt 3 }}{3}} \right)$$

$$\eqalign{ & \int_{\pi /6}^{\pi /3} {{{\csc }^3}x} dx \cr & {\text{Rewrite the integrand}} \cr & = \int_{\pi /6}^{\pi /3} {{{\csc }^2}x\csc x} dx \cr & {\text{Integrate by parts, }}\, \cr & {\text{Let }}u = \csc x,{\text{ }}du = - \csc x\cot xdx \cr & dv = {\csc ^2}xdx,{\text{ }}v = - \cot x \cr & \int {udv} = uv - \int {vdu} \cr & \int {{{\csc }^3}x} dx = \left( {\csc x} \right)\left( { - \cot x} \right) - \int {\left( { - \cot x} \right)\left( { - \csc x\cot x} \right)} dx \cr & \int {{{\csc }^3}x} dx = - \csc x\cot x - \int {{{\cot }^2}x\csc x} dx \cr & {\text{Use the pythagorean identity }}1 + {\cot ^2}x = {\csc ^2}x \cr & \int {{{\csc }^3}x} dx = - \csc x\cot x - \int {\left( {{{\csc }^2}x - 1} \right)\csc x} dx \cr & \int {{{\csc }^3}x} dx = - \csc x\cot x - \int {\left( {{{\csc }^3}x - \csc x} \right)} dx \cr & \int {{{\csc }^3}x} dx = - \csc x\cot x - \int {{{\csc }^3}x} dx + \int {\csc x} dx \cr & 2\int {{{\csc }^3}x} dx = - \csc x\cot x + \int {\csc x} dx \cr & {\text{Where }}\int {\csc x} dx = \ln \left| {\csc x - \cot x} \right| + C \cr & 2\int {{{\csc }^3}x} dx = - \csc x\cot x + \ln \left| {\csc x - \cot x} \right| + C \cr & \int {{{\csc }^3}x} dx = - \frac{1}{2}\csc x\cot x + \frac{1}{2}\ln \left| {\csc x - \cot x} \right| + C \cr & {\text{Therefore,}} \cr & \int_{\pi /6}^{\pi /3} {{{\csc }^3}x} dx = \left[ { - \frac{1}{2}\csc x\cot x + \frac{1}{2}\ln \left| {\csc x - \cot x} \right|} \right]_{\pi /6}^{\pi /3} \cr & = - \frac{1}{2}\left[ {\csc \left( {\frac{\pi }{3}} \right)\cot \left( {\frac{\pi }{3}} \right) - \ln \left| {\csc \frac{\pi }{3} - \cot \frac{\pi }{3}} \right|} \right] \cr & + \frac{1}{2}\left[ {\csc \left( {\frac{\pi }{6}} \right)\cot \left( {\frac{\pi }{6}} \right) - \ln \left| {\csc \frac{\pi }{6} - \cot \frac{\pi }{6}} \right|} \right] \cr & {\text{Simplifying}} \cr & = - \frac{1}{2}\left[ {\left( {\frac{{2\sqrt 3 }}{3}} \right)\left( {\frac{{\sqrt 3 }}{3}} \right) - \ln \left| {\frac{{2\sqrt 3 }}{3} - \frac{{\sqrt 3 }}{3}} \right|} \right] \cr & + \frac{1}{2}\left[ {2\sqrt 3 - \ln \left| {2 - \sqrt 3 } \right|} \right] \cr & = - \frac{1}{2}\left[ {\frac{2}{3} - \ln \left| {\frac{{\sqrt 3 }}{3}} \right|} \right] + \frac{1}{2}\left[ {2\sqrt 3 - \ln \left| {2 - \sqrt 3 } \right|} \right] \cr & = - \frac{1}{3} + \frac{1}{2}\ln \left( {\frac{{\sqrt 3 }}{3}} \right) + \sqrt 3 - \frac{1}{2}\ln \left| {2 - \sqrt 3 } \right| \cr & = \sqrt 3 - \frac{1}{3} + \frac{1}{2}\ln \left( {\frac{{3 + 2\sqrt 3 }}{3}} \right) \approx 1.7825 \cr} $$