Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ .
A) $\frac { 1 } { 2 \ln 2 }$
B) $- \frac { 1 } { 2 \ln 2 }$
C)Diverges
D)0

Question

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The integral $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ can be evaluated by recognizing that $\ln \sqrt { x } = \frac{1}{2} \ln x$. Substituting this into the integral gives $\int _ { 2 } ^ { \infty } \frac { 2 } { x \ln x } d x$, which is a known divergent integral because the denominator $\ln x$ grows too slowly to counteract the $\frac{1}{x}$ term as $x$ approaches infinity.

Evaluate ∫2∞1xln⁡xdx\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x∫2∞​xlnx​1​dx

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$