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

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

Question

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

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

Quizplus · Accepted Answer

The integral $\int \sqrt { 4 x ^ { 2 } + 9 } d x$ can be solved using trigonometric substitution or recognizing it as a standard form related to the derivative of the inverse sine function or hyperbolic sine function. The correct answer involves recognizing the form and applying the appropriate integration technique, which leads to the expression involving $\sin ^ { - 1 }$ and the original function under the square root, correctly represented in option A.

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

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