Evaluate the integral.
-$$\int \frac { d x } { x \sqrt { 6 + 4 x } }$$
A) $\frac { 1 } { \sqrt { 6 } } \ln \left| \frac { \sqrt { 6 + 4 x } - \sqrt { 6 } } { \sqrt { 6 + 4 x } + \sqrt { 6 } } \right| + C$
B) $\frac { 1 } { \sqrt { 6 } } \ln \left| \frac { \sqrt { 6 + 4 x } + \sqrt { 6 } } { \sqrt { 6 + 4 x } - \sqrt { 6 } } \right| + C$
C) $2 \sqrt { 6 + 4 x } + 2 \sqrt { 6 } \tan ^ { - 1 } \sqrt { \frac { 6 + 4 x } { 6 } } + C$
D) $\frac { 1 } { \sqrt { 6 } } \tan ^ { - 1 } \sqrt { \frac { 6 + 4 x } { 6 } } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \frac { d x } { x \sqrt { 6 + 4 x } }$, we can use a substitution method. Let $u = \sqrt{6+4x}$, then differentiate both sides to find $du$ in terms of $dx$, and substitute back into the integral. The solution involves a logarithmic function, and the correct form, after simplifying and back-substituting, matches choice A.

Evaluate the Integral ∫dxx6+4x\int \frac { d x } { x \sqrt { 6 + 4 x } }∫x6+4x​dx​

Evaluate the Integral $\int \frac { d x } { x \sqrt { 6 + 4 x } }$