Question

\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}} dx\]

Answer 1

\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}}dx\]

\[Let, x - \cos x = t\]

\[ \Rightarrow \left( 1 + \sin x \right) = \frac{dt}{dx}\]

\[ \Rightarrow \left( 1 + \sin x \right) dx = dt\]

\[Now, \int\frac{1 + \sin x}{\sqrt{x - \cos x}}dx\]

\[ = \int\frac{dt}{\sqrt{t}}\]

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

\[ = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C\]

\[ = 2\sqrt{t} + C\]

\[ = 2\sqrt{x - \cos x} + C\]