Write a value of + 1+ 2x dx - Vidyadip

Question

Write a value of $\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{1+\sin2\text{x}}}\text{ dx}$

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Answer

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

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