Evaluate:

Question:

Evaluate: $\int \sin x \sqrt{1-\cos 2 x} d x$

Solution:

$\Rightarrow 2 \sin ^{2} x=1-\cos 2 x$

We can substitute the above result in the given equation

$\therefore$ The given equation becomes

$\Rightarrow \int \sin x \sqrt{2 \sin ^{2} x}$

$\Rightarrow \int \sqrt{2} \sin ^{2} x$

$\sin ^{2} x=\frac{1-\cos 2 x}{2}$

$\Rightarrow \frac{\sqrt{2}}{2} \int 1-\cos 2 x d x$

$\Rightarrow \frac{1}{\sqrt{2}} \int d x-\frac{1}{\sqrt{2}} \int \cos 2 x d x$

$\Rightarrow \frac{x}{\sqrt{2}}-\frac{1}{2 \sqrt{2}} \sin (2 x)+c$

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