# Chapter 5: Integrals - Practice Exercises - Page 308: 44

$$\frac{2}{3}{\left( {1 + \sec \theta } \right)^{3/2}} + C$$

#### Work Step by Step

\eqalign{ & \int {\left( {\sec \theta \tan \theta } \right)} \sqrt {1 + \sec \theta } d\theta \cr & {\text{use the substitution method }} \cr & u = 1 + \sec \theta,\,\,\,\,du = \sec \theta \tan \theta d\theta,\,\,\,\,\,d\theta = \frac{{du}}{{\sec \theta \tan \theta }} \cr & {\text{Then}}{\text{,}} \cr & \int {\left( {\sec \theta \tan \theta } \right)} \sqrt {1 + \sec \theta } d\theta = \int {\left( {\sec \theta \tan \theta } \right)} \sqrt u \left( {\frac{{du}}{{\sec \theta \tan \theta }}} \right) \cr & = \int {\sqrt u } du \cr & = \int {{u^{1/2}}} du \cr & {\text{integrate}} \cr & = \frac{{{u^{3/2}}}}{{3/2}} + C \cr & = \frac{2}{3}{u^{3/2}} + C \cr & {\text{write in terms of }}\theta;{\text{ replace }}u = 1 + \sec \theta \cr & = \frac{2}{3}{\left( {1 + \sec \theta } \right)^{3/2}} + C \cr}

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