Chapter 7 - Principles Of Integral Evaluation - 7.4 Trigonometric Substitutions - Exercises Set 7.4 - Page 513: 25

$$\frac{{\sqrt 3 - 5}}{{18}} + \frac{{2\sqrt 3 }}{{27}}$$

$$\eqalign{ & \int_1^3 {\frac{{dx}}{{{x^4}\sqrt {{x^2} + 3} }}} \cr & {\text{substitute }}x = \sqrt 3 \tan \theta ,{\text{ }}\,\,\,dx = \sqrt 3 {\sec ^2}\theta d\theta \,\,\,\,\theta = {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt 3 }}} \right) \cr & \,\,\,x = 3 \to \theta = {\tan ^{ - 1}}\left( {\frac{3}{{\sqrt 3 }}} \right) = \frac{\pi }{3} \cr & \,\,\,x = 1 \to \theta = {\tan ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) = \frac{\pi }{6} \cr & \cr & {\text{write the integral in terms of }}\theta \cr & \int_1^3 {\frac{{dx}}{{{x^4}\sqrt {{x^2} + 3} }}} = \int_{\pi /6}^{\pi /3} {\frac{{\sqrt 3 {{\sec }^2}\theta }}{{9{{\tan }^4}\theta \sqrt {3{{\tan }^2}\theta + 3} }}d\theta } \cr & = \int_{\pi /6}^{\pi /3} {\frac{{{{\sec }^2}\theta }}{{9{{\tan }^4}\theta \sqrt {{{\tan }^2}\theta + 1} }}d\theta } \cr & \cr & {\text{use the pythagorean identity }}{\tan ^2} + 1 = {\sec ^2}\theta \cr & = \int_{\pi /6}^{\pi /3} {\frac{{{{\sec }^2}\theta }}{{9{{\tan }^4}\theta \sqrt {{{\sec }^2}\theta } }}d\theta } \cr & = \frac{1}{9}\int_{\pi /6}^{\pi /3} {\frac{{\sec \theta }}{{{{\tan }^4}\theta }}d\theta } \cr & = \frac{1}{9}\int_{\pi /6}^{\pi /3} {\frac{{{{\cos }^3}\theta }}{{{{\sin }^4}\theta }}d\theta } \cr & = \frac{1}{9}\int_{\pi /6}^{\pi /3} {{{\left( {\sin \theta } \right)}^{ - 4}}{{\cos }^2}\theta \cos \theta d\theta } \cr & = \frac{1}{9}\int_{\pi /6}^{\pi /3} {{{\left( {\sin \theta } \right)}^{ - 4}}\left( {1 - {{\sin }^2}\theta } \right)\cos \theta d\theta } \cr & = \frac{1}{9}\int_{\pi /6}^{\pi /3} {\left( {1 - {{\left( {\sin \theta } \right)}^{ - 2}}} \right)\cos \theta d\theta } \cr & {\text{Integrate}} \cr & = \frac{1}{9}\left[ {\sin \theta - \frac{{{{\left( {\sin \theta } \right)}^{ - 1}}}}{{ - 1}}} \right]_{\pi /6}^{\pi /3} \cr & = \frac{1}{9}\left[ {\sin \theta + \csc \theta } \right]_{\pi /6}^{\pi /3} \cr & \cr & {\text{evaluate the limits}} \cr & = \frac{1}{9}\left[ {\sin \left( {\frac{\pi }{3}} \right) + \csc \left( {\frac{\pi }{3}} \right)} \right] - \frac{1}{9}\left[ {\sin \left( {\frac{\pi }{6}} \right) + \csc \left( {\frac{\pi }{6}} \right)} \right] \cr & = \frac{1}{9}\left[ {\frac{{\sqrt 3 }}{2} + \frac{{2\sqrt 3 }}{3}} \right] - \frac{1}{9}\left[ {\frac{1}{2} + 2} \right] \cr & = \frac{{\sqrt 3 }}{{18}} + \frac{{2\sqrt 3 }}{{27}} - \frac{5}{{18}} \cr & = \frac{{\sqrt 3 - 5}}{{18}} + \frac{{2\sqrt 3 }}{{27}} \cr} $$