Evaluate the integral in terms of natural logarithms.
-$$\int _ { 3 } ^ { 18 } \frac { d x } { x \sqrt { x ^ { 2 } + 9 } }$$
A) $\frac { 1 } { 3 } \ln \left( \frac { 6 ( 1 + \sqrt { 2 } ) } { 1 + \sqrt { 37 } } \right)$
B) $- \ln \frac { 1 + \sqrt { 37 } } { 6 }$
C) $\frac { 1 } { 3 } \ln \left( \frac { 1 + \sqrt { 2 } } { 1 + \sqrt { 35 } } \right)$
D) $- \frac { 1 } { 3 } \ln \left( \frac { 1 + \sqrt { 2 } } { 1 + \sqrt { 37 } } \right)$

Question

Quizplus · Accepted Answer

To solve the integral $\int _ { 3 } ^ { 18 } \frac { d x } { x \sqrt { x ^ { 2 } + 9 } }$, we can use a substitution method. Let $u = x^2 + 9$, then $du = 2x dx$. The limits of integration change accordingly, and after simplifying, we integrate with respect to $u$. The result involves the natural logarithm of a function of $u$, which, when evaluated at the new limits and simplified, matches the expression given in option A.

Evaluate the Integral in Terms of Natural Logarithms ∫318dxxx2+9\int _ { 3 } ^ { 18 } \frac { d x } { x \sqrt { x ^ { 2 } + 9 } }∫318​xx2+9​dx​

Evaluate the Integral in Terms of Natural Logarithms $\int _ { 3 } ^ { 18 } \frac { d x } { x \sqrt { x ^ { 2 } + 9 } }$