Question

\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]

Answer 1

\[\int\frac{\tan x}{\sqrt{\cos x}}dx\]
\[ \Rightarrow \int\frac{\sin x}{\cos x \sqrt{\cos x}} dx\]
\[ \Rightarrow \int\frac{\sin x}{\cos {}^\frac{3}{2} x}dx\]
\[Let \cos x = t\]
\[ \Rightarrow - \text{sin x dx }= dt\]
\[ \Rightarrow \sin x = - \frac{dt}{dx}\]
\[Now, \int\frac{\sin x}{\cos {}^\frac{3}{2} x}dx\]

\[ = \int - \frac{1}{t^\frac{3}{2}}dt\]

 

\[ = - \int t^{- \frac{3}{2}} dt\]

 

\[ = - \left[ \frac{t^{- \frac{3}{2} + 1}}{\frac{- 3}{2} + 1} \right] + C\]
\[ = \frac{2}{\sqrt{t}} + C\]
\[ = \frac{2}{\sqrt{\cos x}} + C\]