Question

\[\int\sin x\sqrt{1 + \cos 2x} dx\]

Answer 1

` ∫  sin   x  \sqrt{ 1 + \text{cos 2 x dx} `
`  =  ∫  sin   x . \sqrt{  2cos ^2 x}   dx `  ` [∴  1 + cos 2x = 2 cos^2 X]`
\[ = \sqrt{2}\int\text{sin x }\text{cos x dx}\]
\[ = \frac{\sqrt{2}}{2}\int\text{2 }\text{sin x } \text{cos x dx}\]
\[ = \frac{1}{\sqrt{2}}\int\text{sin 2x dx}\]
\[ = \frac{1}{\sqrt{2}}\left[ \frac{- \cos 2x}{2} \right] + C\]
\[ = \frac{- 1}{2\sqrt{2}}\cos  2x + C\]