Question

${\int }_{\pi /6}^{\pi /4}\frac{1}{\sqrt{\cos x{\sin }^{3}x}}dx$ is equal to

• A. 1
• B. 1/3
• C. -2
• D. none of these

Answer 1

Answer: (D)

Let $I=\underset{\pi /6}{\overset{\pi /4}{\int }}\frac{dx}{\sqrt{cosxsi{n}^{3}x}}$
$=\underset{\pi /6}{\overset{\pi /4}{\int }}\frac{se{c}^{2}x}{\sqrt{ta{n}^{3}x}}dx$
$={\left|\frac{{\left(tanx\right)}^{-1/2}}{-1/2}\right|}_{\pi /6}^{\pi /4}$
$=-2\left[\left(1\right)-{\left(\frac{1}{\sqrt{3}}\right)}^{-1/2}\right]$
$=-2\left[1-{\left(\sqrt{3}\right)}^{1/2}\right]$
$=2{\left(3\right)}^{1/4}-2$