The integral $\int \sqrt { 16 x ^ { 2 } + 9 } d x$ is

A) $\frac { - 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 2 x \sqrt { 16 x ^ { 2 } + 9 } } { 4 } + C$
B) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) - 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
C) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
D) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) - 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
E) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$

Question

The integral $\int \sqrt { 16 x ^ { 2 } + 9 } d x$ is

A) $\frac { - 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 2 x \sqrt { 16 x ^ { 2 } + 9 } } { 4 } + C$
B) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) - 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
C) $- \frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
D) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) - 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$
E) $\frac { 9 \sinh ^ { - 1 } \left( \frac { 4 x } { 3 } \right) + 4 x \sqrt { 16 x ^ { 2 } + 9 } } { 8 } + C$

Quizplus · Accepted Answer

The integral involves a square root of a quadratic expression, which suggests a trigonometric substitution or recognizing it as a form that results in a hyperbolic sine function, $\sinh^{-1}$. The correct answer involves correctly applying the integration technique and simplifying the expression, which matches option E, where the terms inside the integral are correctly integrated and simplified.

The Integral ∫16x2+9dx\int \sqrt { 16 x ^ { 2 } + 9 } d x∫16x2+9​dx

The Integral $\int \sqrt { 16 x ^ { 2 } + 9 } d x$