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Question Number 137315 by liberty last updated on 01/Apr/21

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

Answered by rs4089 last updated on 01/Apr/21

$${\int }_0^{\pi /4}\sqrt{{cos}^3\left(2x\right)}dx$$$$2x=t\Rightarrow 2dx=dt$$$${\int }_0^{\pi /2}\frac{1}{2}{cos}^{\frac{3}{2}}\left(t\right).dt$$$$\frac{1}{2}\frac{1}{2}\frac{\mathrm{\Gamma }\left(\frac{5}{4}\right)\mathrm{\Gamma }\left(\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{7}{4}\right)}$$

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