# Chapter 8: Techniques of Integration - Section 8.2 - Integration by Parts - Exercises 8.2 - Page 455: 32

$$2\sin \sqrt x + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{{\cos \sqrt x }}{{\sqrt x }}} dx \cr & {\text{Integrate by substitution method}} \cr & {\text{Let }}u = \sqrt x,\,\,\,\,du = \frac{1}{{2\sqrt x }}dx,\,\,\,\,dx = 2\sqrt x du \cr & {\text{Then}}{\text{,}} \cr & \int {\frac{{\cos \sqrt x }}{{\sqrt x }}} dx = \int {\frac{{\cos u}}{{\sqrt x }}} \left( {2\sqrt x du} \right) \cr & {\text{Cancel common factor }}\sqrt x \cr & = \int {\frac{{\cos u}}{{}}} \left( {2du} \right) \cr & = 2\int {\cos u} du \cr & {\text{integrating}} \cr & = 2\sin u + C \cr & {\text{replacing }}u = \sqrt x \cr & = 2\sin \sqrt x + C \cr}

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