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0-pi-4-cos-3-2x-dx-




Question Number 137315 by liberty last updated on 01/Apr/21
∫_0 ^( π/4)  (√(cos^3 (2x))) dx ?
$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \:\sqrt{\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{2x}\right)}\:\mathrm{dx}\:? \\ $$
Answered by rs4089 last updated on 01/Apr/21
∫_0 ^( π/4) (√(cos^3 (2x) )) dx  2x=t ⇒2dx=dt  ∫_0 ^( π/2) (1/2)cos^(3/2) (t).dt  (1/2)(1/2)((Γ((5/4))Γ((1/2)))/(Γ((7/4))))
$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \sqrt{{cos}^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:}\:{dx} \\ $$$$\mathrm{2}{x}={t}\:\Rightarrow\mathrm{2}{dx}={dt} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}{cos}^{\frac{\mathrm{3}}{\mathrm{2}}} \left({t}\right).{dt} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{1}}{\mathrm{2}}\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{7}}{\mathrm{4}}\right)} \\ $$

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