Evaluate the integral .
E) integral diverges

Question

Evaluate the integral  .
E) integral diverges

Quizplus · Accepted Answer

By using trigonometric substitution, we can set $x = 2	an(	heta)$, which gives $dx = 2\sec^2(	heta) d	heta$ and $\sqrt{4x^2 + 9} = 2\sec(	heta)$. Substituting these expressions and simplifying, the integral becomes $\int \frac{2\sec^2(	heta)}{4\sec^3(	heta)}d	heta = \frac{1}{2}\int\frac{1}{\cos(	heta)}d	heta = \frac{1}{2}\ln|\sec(	heta)+	an(	heta)|$. Substituting back for $x$ and simplifying further, we get $\frac{1}{2}\ln|\sqrt{4x^2+9}+2x| + C$. The definite integral evaluates to $\frac{1}{2}\ln|19+18| - \frac{1}{2}\ln|9+0| = \frac{1}{2}\ln\frac{37}{9}$, which is choice C.

Evaluate the Integral