QuestionUse trigonometric substitution to evaluate $$I=\int \frac{2}{\sqrt{1-4 x^{2}}} d x$$ A. None of the options B. $\frac{1}{4} \sin ^{-1}(2 x)+C$ C. $-\sin ^{-1}(2 x)-3 x+C$ D. $\sin ^{-1}(2 x)+C$ E. $-\sin ^{-1}(2 x)+C$

Studdy Solution
Substitute back $\theta$ in terms of $x$. Since $x = \frac{1}{2}\sin(\theta)$, we get $\sin(\theta) = 2x$ and thus $\theta = \sin^{-1}(2x)$.
$$I = \sin^{-1}(2x) + C$$
The correct answer is:
$$\boxed{D. \sin^{-1}(2x) + C}$$