Evaluate the integral in terms of natural logarithms.
-$$\int _ { 0 } ^ { \sqrt { 2 } / 2 } \frac { 32 \mathrm { dx } } { \sqrt { 1 + 16 \mathrm { x } ^ { 2 } } }$$
A) $\ln 11$
B) $8 \ln ( 2 \sqrt { 2 } + 3 )$
C) $32 \ln ( 2 \sqrt { 2 } + 3 )$
D) $8 \ln ( 2 \sqrt { 2 } + \sqrt { 7 } )$

Question

Quizplus · Accepted Answer

The integral can be solved by recognizing it as a standard form that involves the inverse hyperbolic function $\text{arsinh}(x)$ or the natural logarithm. The substitution $u = 4x$ simplifies the integral to a form where the antiderivative can be directly applied, leading to the answer $8 \ln ( 2 \sqrt { 2 } + 3 )$.

Evaluate the Integral in Terms of Natural Logarithms ∫02/232dx1+16x2\int _ { 0 } ^ { \sqrt { 2 } / 2 } \frac { 32 \mathrm { dx } } { \sqrt { 1 + 16 \mathrm { x } ^ { 2 } } }∫02​/2​1+16x2​32dx​

Evaluate the Integral in Terms of Natural Logarithms $\int _ { 0 } ^ { \sqrt { 2 } / 2 } \frac { 32 \mathrm { dx } } { \sqrt { 1 + 16 \mathrm { x } ^ { 2 } } }$