Evaluate the integral in terms of natural logarithms.
-$$\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }$$
A) 5
B) $10 \ln ( 1 + \sqrt { 2 } )$
C) $\ln ( 5 + \sqrt { 26 } )$
D) $10 \ln ( 5 + \sqrt { 26 } )$

Question

Quizplus · Accepted Answer

First we can substitute $u=\ln x$, then 
$$\int \frac{10dx}{x\sqrt{1+(\ln x)^2}}=\int\frac{10du}{\sqrt{1+u^2}}=10\ln(u+\sqrt{1+u^2})+C$$
where $C$ is the constant of integration. Substituting back $u=\ln x$ and evaluating the integral from 1 to $e^5$ gives:
$$\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }=10\ln(\ln x+\sqrt{1+(\ln x)^2})\bigg|_1^{e^5}=10\ln(5+\sqrt{26})$$
Therefore, the correct choice is $\boxed{	extbf{(D)}}$.

Evaluate the Integral in Terms of Natural Logarithms ∫1e510dxx1+(ln⁡x)2\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }∫1e5​x1+(lnx)2​10dx​

Evaluate the Integral in Terms of Natural Logarithms $\int _ { 1 } ^ { e ^ { 5 } } \frac { 10 d x } { x \sqrt { 1 + ( \ln x ) ^ { 2 } } }$