Integrate the function.
-$$\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }$$
A) $\frac { 2 } { x ^ { 2 } + 3 } + C$
B) $9 \ln \left| \sqrt { 3 + x ^ { 2 } } \right| + C$
C) $9 \ln \left| x + \sqrt { 3 + x ^ { 2 } } \right| + C$
D) $x + \ln \left| 9 + \sqrt { 3 + x ^ { 2 } } \right| + C$

Question

Quizplus · Accepted Answer

Substituting $x = \sqrt{3} 	an 	heta$ gives us $\mathrm{dx} = \frac{3}{\sqrt{3+\mathrm{x}^2}} \mathrm{d} 	heta$, and $\sqrt{3+\mathrm{x}^2} = 3 \sec 	heta$. Hence,
\begin{align*}
\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } } &= \int \frac { 9 \cdot 3 } { 3 \sec 	heta } \cdot \frac{1}{3} \sec 	heta \mathrm{d} 	heta \
&= 9\int \cos 	heta \mathrm{d}	heta \
&= 9\sin	heta \
&= 9\frac{x}{\sqrt{3+x^2}} + C \
&= 9\ln \left| x + \sqrt{3+x^2} ight| + C.
\end{align*}Therefore, the correct choice is $\boxed{	extbf{(C)}}$.

Integrate the Function ∫9dx3+x2\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }∫3+x2​9dx​

Integrate the Function $\int \frac { 9 \mathrm { dx } } { \sqrt { 3 + \mathrm { x } ^ { 2 } } }$