Question

Evaluate: ${\int }_{1/\sqrt{2}}^{\sqrt{3}/2}\frac{1}{\sqrt{1-x^2}}dx$

Answer 1

${\int }_{\frac{1}{\sqrt{2}}}^{\frac{\sqrt{3}}{2}}\frac{dx}{\sqrt{1-x^2}}$

Let $x=\sin \theta \Rightarrow dx=\cos \theta d\theta$

When $x=\frac{\sqrt{3}}{2}\Rightarrow \theta =\frac{\pi }{3}$

When $x=\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4}$

$={\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}\frac{dx}{\sqrt{1-{\sin }^2\theta }}$

$={\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}\frac{1}{\cos \theta }\cos \theta d\theta$

$={\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}d\theta$

$={\left[\theta \right]}_{\frac{\pi }{4}}^{\frac{\pi }{3}}$

$=\frac{\pi }{3}-\frac{\pi }{4}=\frac{4\pi -3\pi }{12}=\frac{\pi }{12}$