# Chapter 8 - Further Techniques and Applications of Integration - 8.1 Integration by Parts - 8.1 Exercises - Page 433: 23

$$16\ln \left| {x + \sqrt {{x^2} + 16} } \right| + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{{16}}{{\sqrt {{x^2} + 16} }}} dx \cr & or \cr & = 16\int {\frac{1}{{\sqrt {{x^2} + {{\left( 4 \right)}^2}} }}} dx \cr & {\text{integrate by tables using the formulas on the apendix D for this book}} \cr & {\text{using the formula 5}}:\,\,\,\,\,\,\,\int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = \ln \left| {x + \sqrt {{x^2} + {a^2}} } \right|} + C \cr & {\text{setting }}a = 4{\text{ then}} \cr & = 16\int {\frac{1}{{\sqrt {{x^2} + 16} }}} dx = 16\left( {\ln \left| {x + \sqrt {{x^2} + {{\left( 4 \right)}^2}} } \right|} \right) + C \cr & {\text{simplifying}} \cr & \int {\frac{{16}}{{\sqrt {{x^2} + 16} }}} dx = 16\ln \left| {x + \sqrt {{x^2} + 16} } \right| + C \cr}

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