Prove

Question:

$\sqrt{\sin 2 x} \cos 2 x$

Solution:

Let $\sin 2 x=t$

$\therefore 2 \cos 2 x d x=d t$

$\Rightarrow \int \sqrt{\sin 2 x} \cos 2 x d x=\frac{1}{2} \int \sqrt{t} d t$

$=\frac{1}{2}\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+\mathrm{C}$

$=\frac{1}{3} t^{\frac{3}{2}}+\mathrm{C}$

$=\frac{1}{3}(\sin 2 x)^{\frac{3}{2}}+\mathrm{C}$

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