Find the indefinite integral. $$\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x$$
A)$- \frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$
B)$\frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$
C)$- \frac { x } { 8 \sqrt { x ^ { 2 } + 8 } } + C$
D)$\frac { x ^ { 2 } } { 8 \sqrt { x ^ { 2 } + 8 } } + C$
E) $8 x \sqrt { x ^ { 2 } + 8 } + C$

Question

Quizplus · Accepted Answer

This integral can be solved using a trigonometric substitution, but a quicker way to see the correct answer is by recognizing the derivative of the square root function in the denominator. The derivative of $\sqrt{x^2 + 8}$ with respect to $x$ is $\frac{x}{\sqrt{x^2 + 8}}$, which closely resembles the options given. Among the options, only option B has the correct form $\frac{x}{8\sqrt{x^2 + 8}} + C$, which is the result of integrating the given function.

Find the Indefinite Integral ∫1(x2+8)3/2dx\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x∫(x2+8)3/21​dx

Find the Indefinite Integral $\int \frac { 1 } { \left( x ^ { 2 } + 8 \right) ^ { 3 / 2 } } d x$