Find the integral using an appropriate trigonometric substitution.

Question

Quizplus · Accepted Answer

The integral involves the square root of the quadratic expression $4x^2+5$. This suggests that we can use a trigonometric substitution $x=\frac{\sqrt{5}}{2}	an	heta$. Then, we have 
\begin{align*}
4x^2+5 &= 4\left(\frac{\sqrt{5}}{2}	an	hetaight)^2+5 \
&= 5\sec^2	heta,
\end{align*}
and $dx = \frac{\sqrt{5}}{2}\sec^2	heta d	heta$. Substituting these expressions into the given integral, we get 
\begin{align*}
\int\frac{\sqrt{4x^2+5}}{x}dx &= \int \frac{\sqrt{5\sec^2	heta}}{\frac{\sqrt{5}}{2}	an	heta}\cdot\frac{\sqrt{5}}{2}\sec^2	heta d	heta \
&= \int \frac{5}{2}\cdot\frac{\sec^3	heta}{	an	heta}d	heta\
&= \int \frac{5}{2}\sec^2	heta d	heta \
&= \frac{5}{2}	an	heta + C \
&= \frac{5}{2}	an\left(\arctan\frac{2x}{\sqrt{5}}ight) + C \
&= \frac{5}{2}\cdot\frac{2x}{\sqrt{5}}+C \
&= x\sqrt{5}+C.
\end{align*}
Hence, the correct choice is $\boxed{	ext{(B)}}$.

Find the Integral Using an Appropriate Trigonometric Substitution