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Question
\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}} dx\]
Sum
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Solution
\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}}dx\]
\[Let, x - \cos x = t\]
\[ \Rightarrow \left( 1 + \sin x \right) = \frac{dt}{dx}\]
\[ \Rightarrow \left( 1 + \sin x \right) dx = dt\]
\[Now, \int\frac{1 + \sin x}{\sqrt{x - \cos x}}dx\]
\[ = \int\frac{dt}{\sqrt{t}}\]
\[ = \int t^{- \frac{1}{2}} dt\]
\[ = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C\]
\[ = 2\sqrt{t} + C\]
\[ = 2\sqrt{x - \cos x} + C\]
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