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x-1-x-3-dx-




Question Number 152660 by Tawa11 last updated on 30/Aug/21
∫ (x/( (√(1   +   x^3 )))) dx
$$\int\:\frac{\mathrm{x}}{\:\sqrt{\mathrm{1}\:\:\:+\:\:\:\mathrm{x}^{\mathrm{3}} }}\:\mathrm{dx} \\ $$
Answered by Olaf_Thorendsen last updated on 31/Aug/21
(1+u)^(−(1/2))  = Σ_(n=0) ^∞ (−1)^n (((2n)!)/(2^(2n) (n!)^2 ))u^n   x(1+x^3 )^(−(1/2))  = Σ_(n=0) ^∞ (−1)^n (((2n)!)/(2^(2n) (n!)^2 ))x^(3n+1)   ∫x(1+x^3 )^(−(1/2)) dx = Σ_(n=0) ^∞ (−1)^n (((2n)!)/((3n+2)2^(2n) (n!)^2 ))x^(3n+2) +C
$$\left(\mathrm{1}+{u}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{u}^{{n}} \\ $$$${x}\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{x}^{\mathrm{3}{n}+\mathrm{1}} \\ $$$$\int{x}\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{−\frac{\mathrm{1}}{\mathrm{2}}} {dx}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{\left(\mathrm{2}{n}\right)!}{\left(\mathrm{3}{n}+\mathrm{2}\right)\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{x}^{\mathrm{3}{n}+\mathrm{2}} +\mathrm{C} \\ $$
Commented by Tawa11 last updated on 31/Aug/21
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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