Evaluate the following: + 1+ 2x dx - Vidyadip

Question

Evaluate the following:
$\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{1+\sin2\text{x}}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{1+\sin2\text{x}}}\text{dx}$ $=\int\frac{(\sin\text{x}+\cos\text{x})}{\sqrt{\sin^2\text{x}+\cos^2\text{x}+2\sin\text{x}\cos\text{x}}}\text{dx}$
$=\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{(\sin\text{x}+\cos\text{x})^2}}\text{dx}$ $=\int1\text{dx}=\text{x}+\text{C}$

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