Question

$\int \frac{1}{\sqrt{1-(2-\sqrt{3}x{)}^2}}dx$ is equal to
(where $C$ is constant of integration)

• A. ${\sin }^{-1}(2-\sqrt{3}x)+c$
• B. $\frac{1}{\sqrt{3}}{\sin }^{-1}(2-\sqrt{3}x)+c$
• C. $\frac{1}{\sqrt{3}}{\sin }^{-1}(\sqrt{3}x-2)+c$
• D. $\frac{1}{\sqrt{3}}{\sin }^{-1}(\sqrt{3}x+2)+c$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}{\sin }^{-1}(\sqrt{3}x-2)+c$

Let, $2-\sqrt{3}x=t$
$\Rightarrow -\sqrt{3}dx=dt\Rightarrow dx=-\frac{1}{\sqrt{3}}dt$
$=\frac{-1}{\sqrt{3}}\int \frac{1}{\sqrt{1-t^2}}dt=-\frac{1}{\sqrt{3}}{\sin }^{-1}t+c=-\frac{1}{\sqrt{3}}{\sin }^{-1}(2-\sqrt{3}x)+c=\frac{1}{\sqrt{3}}{\sin }^{-1}(\sqrt{3}x-2)+c$