let-A-n-0-pi-2-cos-n-xdx-1-calculate-A-0-A-2-and-A-3-2-calculate-A-n-interms-of-n-3-find-0-pi-2-cos-8-xdx-

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Question Number 65770 by mathmax by abdo last updated on 03/Aug/19

$$let{A}_n={\int }_0^{\frac{\pi }{2}}{cos}^{n}xdx$$$$1)calculate{A}_0,A_{2}and{A}_3$$$$2)calculate{A}_{n}intermsofn$$$$3)find{\int }_0^{\frac{\pi }{2}}{cos}^{8}xdx$$

Commented by ~ À ® @ 237 ~ last updated on 04/Aug/19

$$A_0={\left[x\right]}_{0_{}}^{\frac{\pi }{2}}=\frac{\pi }{2}$$$$2A_n=2{\int }_0^{\frac{\pi }{2}}{\left(sinx\right)}^{2\left(\frac{1}{2}\right)-1}{\left(cosx\right)}^{2\left(\frac{n+1}{2}\right)-1}dx=B(\frac{1}{2}.\frac{n+1}{2})$$$$Then{A}_n=\frac{1}{2}.\frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{n+1}{2}\right)}{\mathrm{\Gamma }\left(\frac{n+2}{2}\right)}$$$$whenusingtheLagrangeformula\underset{k=1}{\prod ^{p-1}}\mathrm{\Gamma }(z+\frac{k}{p})={\left(2\pi \right)}^{\frac{p-1}{2}}{p}^{\frac{1}{2}-pz}\mathrm{\Gamma }\left(pz\right)$$$$\mathrm{\Gamma }(n+\frac{1}{2})=\sqrt{\pi }\frac{\mathrm{\Gamma }\left(2n\right)}{2^{2n-1}}\mathrm{\forall }n>1$$$$sowecanfind$$$$A_{2n}=\frac{\sqrt{\pi }}{2}.\frac{\frac{\sqrt{\pi }\mathrm{\Gamma }\left(2n\right)}{2^{2n-1}}}{\mathrm{\Gamma }(n+1)}=\frac{(2n-1)!}{2^{2n}n!}\pi$$$$A_{2n+1}=\frac{\sqrt{\pi }}{2}.\frac{\mathrm{\Gamma }(n+1)}{\sqrt{\pi }.\frac{\mathrm{\Gamma }(2n+2)}{2^{2n+1}}}=\frac{2^{2n}.n!}{(2n+1)!}$$$$So{A}_1=1A_3=\frac{2^2.1!}{3!}{A}_8=\frac{7!\pi }{2^{8}4!}$$

Commented by mathmax by abdo last updated on 04/Aug/19

$$thankyousir.$$

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