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dx-1-x-4-




Question Number 140339 by mohammad17 last updated on 06/May/21
∫(dx/( (√(1+x^4 ))))
$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$
Answered by Dwaipayan Shikari last updated on 06/May/21
∫(dx/( (√(1+x^4 ))))=Σ_(n≥0) ∫((((1/2))_n )/(n!))(−x^4 )^n   =xΣ_(n≥0) ((((1/2))_n ((1/4))_n )/(n!((5/4))_n ))(−x^4 )^n =x _2 F_1 ((1/2),(1/4),(5/4);−x^4 )+C
$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}=\underset{{n}\geqslant\mathrm{0}} {\sum}\int\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} }{{n}!}\left(−{x}^{\mathrm{4}} \right)^{{n}} \\ $$$$={x}\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)_{{n}} }{{n}!\left(\frac{\mathrm{5}}{\mathrm{4}}\right)_{{n}} }\left(−{x}^{\mathrm{4}} \right)^{{n}} ={x}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{4}},\frac{\mathrm{5}}{\mathrm{4}};−{x}^{\mathrm{4}} \right)+{C} \\ $$

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