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

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

Question

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

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

Quizplus · Accepted Answer

To solve the integral $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 4 } } d x$, one can use substitution and integration techniques that involve recognizing the derivative of the function inside the square root in the denominator. The correct answer involves both a term with $x$ times the square root of $x^2 + 4$ and a term involving the inverse hyperbolic sine function, $\sinh^{-1}\left(\frac{x}{2}\right)$, with a specific coefficient. The correct expression is obtained by integrating the function and simplifying the result, which matches option D.

The Integral ∫x2x2+4dx\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 4 } } d x∫x2+4​x2​dx

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