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Question Number 137345 by liberty last updated on 01/Apr/21

$$\ell ={\int }_0^{\pi /4}\sqrt{\cos {}^3\left(2x\right)}\cos xdx=?$$

Answered by MJS_new last updated on 01/Apr/21

$$\int {\cos }^{3/2}2x\cos xdx=$$$$[t=\frac{\sqrt{2}\sin x}{{\cos }^{1/2}2x}\to dx=\frac{{\cos }^{3/2}2x}{\sqrt{2}\cos x}]$$$$=\frac{1}{\sqrt{2}}\int \frac{dt}{{(t^2+1)}^3}=$$$$\left[\mathrm{Ostrogradski}\right]$$$$=\frac{\sqrt{2}(3t^2+5)t}{16(t^2+1)}+\frac{3\sqrt{2}}{16}\int \frac{dt}{t^2+1}=$$$$=\frac{\sqrt{2}(3t^2+5)t}{16(t^2+1)}+\frac{3\sqrt{2}}{16}\arctan t=$$$$=\frac{\sin x(3+\cos 2x){\cos }^{1/2}2x}{8}+\frac{3\sqrt{2}\arcsin \left(\sqrt{2}\sin x\right)}{16}+C$$$$\Rightarrow \mathrm{answer}\mathrm{is}\frac{3\sqrt{2}\pi }{32}$$

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