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x-3-1-1-3-dx-solution-

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Question Number 193426 by 073 last updated on 13/Jun/23
∫((x^3 +1))^(1/3) dx=? solution?
$$\int\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}\mathrm{dx}=? \\ $$$$\mathrm{solution}? \\ $$
Answered by witcher3 last updated on 15/Jun/23
x^3 =t ∫(x^3 +1)^(1/3) dx=f(x) (1/3)∫(t+1)^(1/3) t^(−(2/3)) dt (t+1)^(1/3) =1+Σ_(k≥1) ((Π_(j=1) ^k ((4/3)−j))/((k)!))t^k (1/3)∫t^(−(2/3)) +Σ_(k≥1) (((−1)^k Γ(k−(1/3)))/(Γ(k+1)Γ(−(1/3))))t^(k−(2/3)) dt =t^(1/3) +(1/3)Σ_(k≥1) ((Γ(k−(1/3))(−1)^k )/(Γ(−(1/3))(k+(1/3))))(t^(k+(1/3)) /(k!))+c t^(1/3) (1+(1/3)Σ_(k≥1) ((Γ(k−(1/3)))/(Γ(−(1/3)))).((Γ(k+(1/3)))/(Γ(k+(4/3)))).(((−t)^k )/(k!)))+c (1/3)=((Γ((4/3)))/(Γ((1/3)))) ⇔t^(1/3) (1+Σ_(k≥1) ((Γ(k−(1/3)))/(Γ(−(1/3)))).(((Γ(k+(1/3)))/(Γ((1/3))))/((Γ(k+(4/3)))/(Γ((4/3))))).(((−t)^k )/(k!)))+c (a)_k =((Γ(a+k))/(Γ(a))) t^(1/3) (1+Σ_(k≥1) (((−(1/3))_k ((1/3))_k )/(((4/3))_k )).(((−t)^k )/(k!)))+c t^(1/3) _2 F_1 (−(1/3),(1/3);(4/3),−t)+c f(x)=x_2 F_1 (−(1/3),(1/3);(4/3),−x^3 )+c,c∈R
$$\mathrm{x}^{\mathrm{3}} =\mathrm{t} \\ $$$$\int\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{dx}=\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\int\left(\mathrm{t}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{dt} \\ $$$$\left(\mathrm{t}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} =\mathrm{1}+\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{k}} {\prod}}\left(\frac{\mathrm{4}}{\mathrm{3}}−\mathrm{j}\right)}{\left(\mathrm{k}\right)!}\mathrm{t}^{\mathrm{k}} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\int\mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} +\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} \Gamma\left(\mathrm{k}−\frac{\mathrm{1}}{\mathrm{3}}\right)}{\Gamma\left(\mathrm{k}+\mathrm{1}\right)\Gamma\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)}\mathrm{t}^{\mathrm{k}−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{dt} \\ $$$$=\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{3}}} +\frac{\mathrm{1}}{\mathrm{3}}\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\Gamma\left(\mathrm{k}−\frac{\mathrm{1}}{\mathrm{3}}\right)\left(−\mathrm{1}\right)^{\mathrm{k}} }{\Gamma\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)\left(\mathrm{k}+\frac{\mathrm{1}}{\mathrm{3}}\right)}\frac{\mathrm{t}^{\mathrm{k}+\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{k}!}+\mathrm{c} \\ $$$$\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\Gamma\left(\mathrm{k}−\frac{\mathrm{1}}{\mathrm{3}}\right)}{\Gamma\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)}.\frac{\Gamma\left(\mathrm{k}+\frac{\mathrm{1}}{\mathrm{3}}\right)}{\Gamma\left(\mathrm{k}+\frac{\mathrm{4}}{\mathrm{3}}\right)}.\frac{\left(−\mathrm{t}\right)^{\mathrm{k}} }{\mathrm{k}!}\right)+\mathrm{c} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}=\frac{\Gamma\left(\frac{\mathrm{4}}{\mathrm{3}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$$$\Leftrightarrow\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}+\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\Gamma\left(\mathrm{k}−\frac{\mathrm{1}}{\mathrm{3}}\right)}{\Gamma\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)}.\frac{\frac{\Gamma\left(\mathrm{k}+\frac{\mathrm{1}}{\mathrm{3}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)}}{\frac{\Gamma\left(\mathrm{k}+\frac{\mathrm{4}}{\mathrm{3}}\right)}{\Gamma\left(\frac{\mathrm{4}}{\mathrm{3}}\right)}}.\frac{\left(−\mathrm{t}\right)^{\mathrm{k}} }{\mathrm{k}!}\right)+\mathrm{c} \\ $$$$\left(\mathrm{a}\right)_{\mathrm{k}} =\frac{\Gamma\left(\mathrm{a}+\mathrm{k}\right)}{\Gamma\left(\mathrm{a}\right)} \\ $$$$\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}+\underset{\mathrm{k}\geqslant\mathrm{1}} {\sum}\frac{\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)_{\mathrm{k}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)_{\mathrm{k}} }{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)_{\mathrm{k}} }.\frac{\left(−\mathrm{t}\right)^{\mathrm{k}} }{\mathrm{k}!}\right)+\mathrm{c} \\ $$$$\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{3}}} \:_{\mathrm{2}} \mathrm{F}_{\mathrm{1}} \left(−\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{1}}{\mathrm{3}};\frac{\mathrm{4}}{\mathrm{3}},−\mathrm{t}\right)+\mathrm{c} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}_{\mathrm{2}} \mathrm{F}_{\mathrm{1}} \left(−\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{1}}{\mathrm{3}};\frac{\mathrm{4}}{\mathrm{3}},−\mathrm{x}^{\mathrm{3}} \right)+\mathrm{c},\mathrm{c}\in\mathbb{R} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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