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

A) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
B) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) - 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
C) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
D) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) - 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
E) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$

Question

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

A) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
B) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) - 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
C) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
D) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) - 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$
E) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 2 x } { 3 } \right) + 2 x \sqrt { 4 x ^ { 2 } + 9 } } { 4 } + C$

Quizplus · Accepted Answer

The integral $\int \sqrt { 4 x ^ { 2 } + 9 } d x$ can be solved using a trigonometric substitution, and the correct solution involves the inverse hyperbolic sine function ($\sinh^{-1}$) and the original expression under the square root. The correct answer is the one that correctly combines these elements with the right coefficients and signs.

The Integral ∫4x2+9dx\int \sqrt { 4 x ^ { 2 } + 9 } d x∫4x2+9​dx

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