The integral $\int \frac { x ^ { 2 } } { \sqrt { 9 x ^ { 2 } - 4 } } d x$ is

A) $- \frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
B) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } - 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
C) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } - 18 x \right) - 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
D) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
E) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) - 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$

Question

The integral $\int \frac { x ^ { 2 } } { \sqrt { 9 x ^ { 2 } - 4 } } d x$ is

A) $- \frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
B) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } - 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
C) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } - 18 x \right) - 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
D) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) + 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$
E) $\frac { 4 \ln \left( 6 \sqrt { 9 x ^ { 2 } - 4 } + 18 x \right) - 3 x \sqrt { 9 x ^ { 2 } - 4 } } { 54 } + C$

Quizplus · Accepted Answer

The integral can be solved by making a substitution that simplifies the square root in the denominator, such as $u = 9x^2 - 4$, which leads to a solution involving both a logarithmic and a linear term in $x$. The correct answer must correctly reflect the structure of the integral's solution, including the signs and coefficients of both the logarithmic and the linear terms. Option D correctly combines these elements, indicating the correct manipulation and integration of the given expression.

The Integral ∫x29x2−4dx\int \frac { x ^ { 2 } } { \sqrt { 9 x ^ { 2 } - 4 } } d x∫9x2−4​x2​dx

The Integral $\int \frac { x ^ { 2 } } { \sqrt { 9 x ^ { 2 } - 4 } } d x$