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Question Number 98929 by Dwaipayan Shikari last updated on 17/Jun/20

$$Find\left[\right]the\left[\right]integral\left[\right]of\left[\right]$$$$$$$$\int \frac{dt}{\sqrt{(1+t^{10})}}$$

Commented by maths mind last updated on 17/Jun/20

$$\frac{1}{\sqrt{1+x}}=1+\sum _{n⩾1}\frac{\underset{k=1}{\prod ^n}(\frac{1}{2}-k)}{n!}{x}^n$$$$=1+\sum _{n⩾1}\frac{{(-1)}^n\left(2n\right)!}{4^n.{(n!)}^2}{x}^n$$$$\frac{1}{\sqrt{1+t^{10}}}=1+\sum _{n⩾1}\frac{{(-1)}^n.2n!}{4^n.n!.n!}.x^{10n}$$$$\int \frac{dt}{\sqrt{1+t^{10}}}=x(1+\sum _{n⩾1}\frac{{(-1)}^n\left(2n\right)!}{4^n.{(n!)}^2}.\frac{x^{10n}}{10n+1})+c$$$$2n!=4^n.n!.\underset{k=0}{\prod ^{n-1}}(k+\frac{1}{2})$$$$=x(1+\sum _{n⩾1}\frac{{(-1)}^n.4^n.n!.\underset{k=0}{\prod ^{n-1}}(k+\frac{1}{2})}{4^n.n!.10(n+\frac{1}{10})}.\frac{x^{10n}}{n!})+c$$$$=x(1+\sum _{n⩾1}\frac{\underset{k=0}{\prod ^{n-1}}(k+\frac{1}{2}).\underset{k=0}{\prod ^{n-1}}(k+\frac{1}{10})}{\underset{k=0}{\prod ^{n-1}}(k+\frac{11}{10})}.\frac{{(-x^{10})}^n}{n!})+c$$$$=x{}_2{F}_1(\frac{1}{2},\frac{1}{10};\frac{11}{10};-x^{10})+c$$

Commented by  M±th+et+s last updated on 17/Jun/20

$$welldone.icheckedit{it}^{\prime }sniceandcorrect$$

Commented by maths mind last updated on 18/Jun/20

$$thankyouSir$$$$icantfindyourpost\sum _n\sum _m\frac{\mathrm{\Gamma }(n+\frac{1}{2}).\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(m\right)}......$$$$canyousendtheIdofquationigotanideathankyou$$

Commented by  M±th+et+s last updated on 18/Jun/20

$$gotoq98434$$

Commented by  M±th+et+s last updated on 18/Jun/20

$$thankyousomuchforsolvingmyproblems$$

Commented by maths mind last updated on 20/Jun/20

$$withePleasurthankyouiwillpostsolution$$$$notyetfinishthisone$$

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