Evaluate the integral.
-$$\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$$
A) $\sqrt { 4 - x ^ { 2 } } - \sin ^ { - 1 } \left( \frac { x } { 2 } \right) + C$
B) $\sqrt { 4 - x ^ { 2 } } - 2 \ln \left| \frac { 2 + \sqrt { 4 - x ^ { 2 } } } { x } \right| + C$
C) $\sqrt { 4 - x ^ { 2 } } + 2 \ln \left| \frac { 2 + \sqrt { 4 - x ^ { 2 } } } { x ^ { 2 } } \right| + C$
D) $- \sin ^ { - 1 } \left( \frac { x } { 2 } \right) - \frac { \sqrt { 4 - x ^ { 2 } } } { x } + C$

Question

Quizplus · Accepted Answer

The integral $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$ can be solved using trigonometric substitution. Let $x = 2\sin(\theta)$, then $dx = 2\cos(\theta)d\theta$, and the integral becomes $\int \frac{2\cos^2(\theta)}{2\sin(\theta)}d\theta$, which simplifies to $\int \frac{\cos(\theta)}{\sin(\theta)}d\theta$. This can be integrated directly or by recognizing it as a standard form, leading to the solution involving a logarithmic function. The correct answer involves a logarithmic expression, which matches choice B.

Evaluate the Integral ∫4−x2xdx\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x∫x4−x2​​dx

Evaluate the Integral $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$