Integrate the function.
-$$\int \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } } d x$$
A) $\frac { 1 } { 5 } \ln \left| \sqrt { x ^ { 2 } + 9 } + x \right| - \sin ^ { - 1 } \frac { x } { 3 } + C$
B) $\ln \left| \sqrt { x ^ { 2 } + 9 } + x \right| + \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x } + C$
C) $\frac { 1 } { 5 } \ln \left| \sqrt { \mathrm { x } ^ { 2 } + 9 } + \mathrm { x } \right| - \frac { \sqrt { \mathrm { x } ^ { 2 } + 9 } } { 5 \mathrm { x } } + \mathrm { C }$
D) $x + \ln \left| \sqrt { x ^ { 2 } + 9 } \right| + \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x } + C$

Question

Quizplus · Accepted Answer

The given integral can be solved using the substitution $x=3\tan\theta$, which gives 
$$\int\frac{\sqrt{x^2+9}}{5x^2}\,dx=\int\frac{\sec^2\theta}{15}\,d\theta=\frac{1}{15}\tan\theta+C=\frac{1}{15}\tan(\sin^{-1}(\frac{x}{3}))+C.$$
Using the identity $\tan(\sin^{-1}(t))=\frac{\sqrt{1-t^2}}{t}$, we get 
$$\int\frac{\sqrt{x^2+9}}{5x^2}\,dx=\frac{1}{15}\cdot\frac{\sqrt{x^2+9}}{x}+\mathrm{C}=\frac{1}{5}\cdot\frac{\sqrt{x^2+9}}{5x}-\frac{1}{5}\cdot\frac{1}{\sqrt{x^2+9}}+\mathrm{C}.$$
Simplifying the expression, we get the answer in the choice C.

Integrate the Function ∫x2+95x2dx\int \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } } d x∫5x2x2+9​​dx

Integrate the Function $\int \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } } d x$