# Chapter 8: Techniques of Integration - Practice Exercises - Page 519: 79

$$\frac{1}{{2\left( {1 + \cos 2\theta } \right)}} + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{{\sin 2\theta d\theta }}{{{{\left( {1 + \cos 2\theta } \right)}^2}}}} \cr & \cr & {\text{Integrate using the substitution method}}{\text{, }} \cr & \,\,\,\,\,u = 1 + \cos 2\theta ,\,\,\,\,\,du = - 2\sin 2\theta d\theta ,\,\,\,\sin 2\theta d\theta = - \frac{1}{2}du \cr & \int {\frac{{\sin 2\theta d\theta }}{{{{\left( {1 + \cos 2\theta } \right)}^2}}}} = \int {\frac{{\left( { - 1/2} \right)du}}{{{u^2}}}} \cr & = - \frac{1}{2}\int {\frac{1}{{{u^2}}}} du \cr & {\text{Integrate}} \cr & = - \frac{1}{2}\left( { - \frac{1}{u}} \right) + C \cr & = \frac{1}{{2u}} + C \cr & \cr & {\text{Write in terms of }}\theta ,{\text{ substitute }}1 + \cos 2\theta {\text{ for }}u \cr & = \frac{1}{{2\left( {1 + \cos 2\theta } \right)}} + C \cr}

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