# Chapter 5: Integrals - Practice Exercises - Page 308: 60

$$3\sqrt 3 - \pi$$

#### Work Step by Step

\eqalign{ & \int_0^\pi {{{\tan }^2}\frac{\theta }{3}} d\theta \cr & {\text{Use the trigonometric identity}}\cr & \tan^2 \theta + 1 = {\sec ^2}\theta \cr & = \int_0^\pi {\left( {{{\sec }^2}\frac{\theta }{3} - 1} \right)} d\theta \cr & {\text{We know that }}\cr &\frac{d}{{dx}}\left[ {\tan x} \right] = {\sec ^2}x\cr &{\text{ Then}}{\text{,}} \cr & = \left( {3\tan \left( {\frac{\theta }{3}} \right) - \theta } \right)_0^\pi \cr & {\text{Evaluate}} \cr & = \left( {3\tan \left( {\frac{\pi }{3}} \right) - \pi } \right) - \left( {3\tan \left( {\frac{0}{3}} \right) - 0} \right) \cr & {\text{Simplifying}} \cr & = 3\left( {\sqrt 3 } \right) - \pi - 0 \cr & = 3\sqrt 3 - \pi \cr}

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