The integral $\int \frac { 1 } { \left( 4 x ^ { 2 } + 9 \right) ^ { \frac { 3 } { 2 } } } d x$ is

A) $\frac { x } { 9 \sqrt { 4 x ^ { 2 } - 9 } } + C$
B) $- \frac { x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$
C) $\frac { x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$
D) $- \frac { x } { 9 \sqrt { 4 x ^ { 2 } - 9 } } + C$
E) $\frac { 2 x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$

Question

The integral $\int \frac { 1 } { \left( 4 x ^ { 2 } + 9 \right) ^ { \frac { 3 } { 2 } } } d x$ is

A) $\frac { x } { 9 \sqrt { 4 x ^ { 2 } - 9 } } + C$
B) $- \frac { x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$
C) $\frac { x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$
D) $- \frac { x } { 9 \sqrt { 4 x ^ { 2 } - 9 } } + C$
E) $\frac { 2 x } { 9 \sqrt { 4 x ^ { 2 } + 9 } } + C$

Quizplus · Accepted Answer

The integral can be solved by recognizing it as a standard form that results from the derivative of the square root function in the denominator. The correct answer, option C, is derived by applying the chain rule in reverse, where the derivative of $4x^2 + 9$ is $8x$, and adjusting the constants to match the integral's form.

The Integral ∫1(4x2+9)32dx\int \frac { 1 } { \left( 4 x ^ { 2 } + 9 \right) ^ { \frac { 3 } { 2 } } } d x∫(4x2+9)23​1​dx

The Integral $\int \frac { 1 } { \left( 4 x ^ { 2 } + 9 \right) ^ { \frac { 3 } { 2 } } } d x$