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Question Number 40619 by math khazana by abdo last updated on 25/Jul/18

$$letf\left(x\right)={\int }_0^{\frac{\pi }{2}}\frac{d\theta }{x+{cos}^2\theta }withx>0.$$$$1)calculatef\left(x\right)and{f}^{{}^{\prime }}\left(x\right)$$$$2)find{f}^{\left(n\right)}\left(x\right)and{f}^{\left(n\right)}\left(0\right)$$$$3)developpfatintegrserie.$$

Commented by abdo mathsup 649 cc last updated on 27/Jul/18

$$1)wehaveprovedthatf\left(x\right)=\frac{\pi \sqrt{2}}{2\sqrt{x}}\Rightarrow$$$$f^{{}^{\prime }}\left(x\right)=\frac{\pi \sqrt{2}}{2}{\left(x^{-\frac{1}{2}}\right)}^{{}^{\prime }}=-\frac{1}{2}\frac{\pi \sqrt{2}}{2}{x}^{-\frac{3}{2}}=-\frac{\pi \sqrt{2}}{4x\sqrt{x}}$$$$2)f^{\left(n\right)}\left(x\right)=\frac{\pi \sqrt{2}}{2}{\left(x^{-\frac{1}{2}}\right)}^{\left(n\right)}letfind$$$${\left(x^p\right)}^{\left(n\right)}withp\in Q$$$${\left(x^p\right)}^{\left(1\right)}={px}^{p-1},{\left(x^p\right)}^{\left(2\right)}=p(p-1)x^{p-2}\Rightarrow$$$${\left(x^p\right)}^{\left(n\right)}=p(p-1)...(p-n+1)x^{p-n}\Rightarrow$$$${\left(x^{-\frac{1}{2}}\right)}^{\left(n\right)}=(-\frac{1}{2})(-\frac{3}{2})...(-\frac{1}{2}-n+1)x^{-\frac{1}{2}-n}$$$$=(-\frac{1}{2})(-\frac{3}{2})...\left(\frac{-1-2n+2}{2}\right)x^{-\frac{1}{2}-n}$$$$=(-\frac{1}{2})(-\frac{3}{2})....(-\frac{2n-1}{2})\frac{1}{x^n\sqrt{x}}\Rightarrow$$$$f^{\left(n\right)}\left(x\right)=\frac{\pi }{\sqrt{2}}(-\frac{1}{2})(-\frac{3}{2})...(-\frac{2n-1}{2})\frac{1}{x^n\sqrt{x}}$$$$f^{\left(n\right)}\left(1\right)=\frac{\pi }{\sqrt{2}}(-\frac{1}{2})(-\frac{3}{2})...(-\frac{2n-1}{2})$$$$3)f\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{\left(n\right)}\left(1\right)}{n!}{(x-1)}^n$$$$=\sum _{n=0}^{\mathrm{\infty }}\frac{\pi }{n!\sqrt{2}}(-\frac{1}{2})(-\frac{3}{2})...(-\frac{2n-1}{2}){(x-1)}^n.$$

Commented by abdo mathsup 649 cc last updated on 27/Jul/18

$$2)theQisfind{f}^{\left(n\right)}\left(x\right)and{f}^{\left(n\right)}\left(1\right)$$$$3)theQisdeveloppfatintegrserieatv\left(1\right).$$

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