Evaluate the following: 1+ dx - Vidyadip

Question

Evaluate the following:
$\int\sqrt{1+\sin\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\sqrt{1+\sin\text{x}}\text{dx}$
$=\int\sqrt{\sin^2\frac{\text{x}}{2}+\cos^2\frac{\text{x}}{2}+2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}\text{dx}$ $\Big[\because\ \sin^2\frac{\text{x}}{2}+\cos^2\frac{\text{x}}{2}=1\Big]$
$=\int\sqrt{\big(\sin\frac{\text{x}}{2}+\cos\frac{\text{x}}{2}\big)^2}\text{dx}$
$=\int\Big(\sin\frac{\text{x}}{2}+\cos\frac{\text{x}}{2}\Big)\text{dx}$
$=-2\cos\frac{\text{x}}{2}+2\sin\frac{\text{x}}{2}+\text{C}$

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