Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.6 Logarithmic And Other Functions Defined By Integrals - Exercises Set 6.6 - Page 459: 4

$$\eqalign{ & \left( {\text{a}} \right)\frac{9}{2} \cr & \left( {\text{b}} \right)9 + \ln 2 \cr & \left( {\text{c}} \right)\ln 2 - 9 \cr & \left( {\text{d}} \right)9 - \ln 2 \cr} $$

$$\eqalign{ & \left( {\text{a}} \right)\int_1^{\sqrt a } {\frac{1}{t}} dt \cr & {\text{Integrate}} \cr & \int_1^{\sqrt a } {\frac{1}{t}} dt = \left[ {\ln t} \right]_1^{\sqrt a } \cr & \left[ {\ln t} \right]_1^{\sqrt a } = \ln \sqrt a - \ln 1 \cr & \left[ {\ln t} \right]_1^{\sqrt a } = \frac{1}{2}\ln a - 0 \cr & {\text{Where }}\ln a = 9 \cr & \left[ {\ln t} \right]_1^{\sqrt a } = \frac{1}{2}\left( 9 \right) - 0 \cr & \left[ {\ln t} \right]_1^{\sqrt a } = \frac{9}{2} \cr & \cr & \left( {\text{b}} \right)\int_1^{2a} {\frac{1}{t}} dt \cr & {\text{Integrate}} \cr & \int_1^{2a} {\frac{1}{t}} dt = \left[ {\ln t} \right]_1^{2a} \cr & \left[ {\ln t} \right]_1^{2a} = \ln 2a - \ln 1 \cr & \left[ {\ln t} \right]_1^{2a} = \ln 2 + \ln a - \ln 1 \cr & \left[ {\ln t} \right]_1^{2a} = \ln 2 + \ln a \cr & {\text{Where }}\ln a = 9 \cr & \left[ {\ln t} \right]_1^{2a} = 9 + \ln 2 \cr & \cr & \left( {\text{c}} \right)\int_1^{2/a} {\frac{1}{t}} dt \cr & {\text{Integrate}} \cr & \int_1^{2/a} {\frac{1}{t}} dt = \left[ {\ln t} \right]_1^{2/a} \cr & \left[ {\ln t} \right]_1^{2/a} = \ln \frac{2}{a} - \ln 1 \cr & \left[ {\ln t} \right]_1^{2/a} = \ln 2 - \ln a \cr & \left[ {\ln t} \right]_1^{2/a} = \ln 2 - \ln a \cr & {\text{Where }}\ln a = 9 \cr & \left[ {\ln t} \right]_1^{2/a} = \ln 2 - 9 \cr & \cr & \left( {\text{d}} \right)\int_1^{{a^3}} {\frac{1}{t}} dt \cr & {\text{Integrate}} \cr & \int_2^a {\frac{1}{t}} dt = \left[ {\ln t} \right]_2^a \cr & \left[ {\ln t} \right]_2^a = \ln a - \ln 2 \cr & {\text{Where }}\ln a = 9 \cr & \left[ {\ln t} \right]_2^a = 9 - \ln 2 \cr} $$