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Question Number 81400 by mind is power last updated on 12/Feb/20

$$HelloNicedayimthinkingofthisoneacloseforme?$$$$\int \sqrt{1+x^p}dx$$$$p\in {\mathbb{R}}_{+},$$$$x\in [0,1[$$

Commented by abdomathmax last updated on 13/Feb/20

$$atformofseriewehave$$$${(1+u)}^{\alpha }=1+\frac{\alpha }{1!}u+\frac{\alpha (\alpha -1)}{2!}{u}^2+...$$$$=1+\sum _{n=1}^{\mathrm{\infty }}\frac{\alpha (\alpha -1)...(\alpha -n+1)}{n!}{u}^n\Rightarrow$$$$\sqrt{1+u}=1+\sum _{n=1}^{\mathrm{\infty }}\frac{\frac{1}{2}(\frac{1}{2}-1)....(\frac{1}{2}-n+1)}{n!}{u}^n\Rightarrow$$$$\sqrt{1+x^p}=1+\sum _{n=1}^{\mathrm{\infty }}\frac{\frac{1}{2}(\frac{1}{2}-1)...(\frac{1}{2}-n+1)}{n!}{x}^{pn}\Rightarrow$$$$\int \sqrt{1+x^p}dx=$$$$1+\sum _{n=1}^{\mathrm{\infty }}\frac{\frac{1}{2}(\frac{1}{2}-1)...(\frac{1}{2}-n+1)}{n!}\times \frac{1}{pn+1}{x}^{pn+1}+C$$

Commented by mind is power last updated on 14/Feb/20

$$=x_2{F}_1(-\frac{1}{2},\frac{1}{p};\frac{1}{p}+1;x^p)+c$$

Commented by mind is power last updated on 14/Feb/20

$$niceSir$$

Answered by behi83417@gmail.com last updated on 12/Feb/20

$${\mathrm{x}}^{\mathrm{p}}={\mathrm{tg}}^2\mathrm{t}\Rightarrow {\mathrm{px}}^{\mathrm{p}-1}\mathrm{dx}=2\mathrm{tgt}(1+{\mathrm{tg}}^2\mathrm{t})\mathrm{dt}$$$$\Rightarrow \mathrm{dx}=\frac{2}{\mathrm{p}}{\mathrm{tg}}^{(\frac{2}{\mathrm{p}}-1)}\mathrm{t}(1+{\mathrm{tg}}^2\mathrm{t})\mathrm{dt}$$$$\Rightarrow \mathrm{I}=\frac{2}{\mathrm{p}}\int {\mathrm{tg}}^{(\frac{2}{\mathrm{p}}-1)}\mathrm{t}.\frac{1}{{\cos }^3\mathrm{t}}\mathrm{dt}=$$$$\stackrel{\mathrm{m}=(\frac{2}{\mathrm{p}}-1)}{=}(\mathrm{m}+1)\int \frac{{\mathrm{tg}}^{\mathrm{m}}\mathrm{t}}{{\cos }^3\mathrm{t}}\mathrm{dt}=$$$$=(\mathrm{m}+1)\int {\sin }^{\mathrm{m}}\mathrm{t}.{\cos }^{-(\mathrm{m}+3)}\mathrm{tdt}=...$$

Commented by mind is power last updated on 12/Feb/20

$$nicesir$$$$TrytoouseHypergeometricFunction$$$$ithinkacloseformebythisonemaybee$$$$$$$$$$

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