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x-x-1-dx-




Question Number 128983 by bramlexs22 last updated on 11/Jan/21
 ∫ ((√x)/(x−1)) dx =?
$$\:\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{x}−\mathrm{1}}\:\mathrm{dx}\:=? \\ $$
Commented by bramlexs22 last updated on 12/Jan/21
Answered by liberty last updated on 12/Jan/21
 let x = ℓ^2    ∫ ((ℓ (2ℓ dℓ))/(ℓ^2 −1)) = ∫ 2 dℓ +∫ ((2 dℓ)/((ℓ+1)(ℓ−1)))   = 2(√x) + ∫ ((1/(ℓ−1))−(1/(ℓ+1)))dℓ   =2(√x) +ln ∣(((√x)−1)/( (√x) +1))∣ + C
$$\:\mathrm{let}\:\mathrm{x}\:=\:\ell^{\mathrm{2}} \\ $$$$\:\int\:\frac{\ell\:\left(\mathrm{2}\ell\:\mathrm{d}\ell\right)}{\ell^{\mathrm{2}} −\mathrm{1}}\:=\:\int\:\mathrm{2}\:\mathrm{d}\ell\:+\int\:\frac{\mathrm{2}\:\mathrm{d}\ell}{\left(\ell+\mathrm{1}\right)\left(\ell−\mathrm{1}\right)} \\ $$$$\:=\:\mathrm{2}\sqrt{{x}}\:+\:\int\:\left(\frac{\mathrm{1}}{\ell−\mathrm{1}}−\frac{\mathrm{1}}{\ell+\mathrm{1}}\right)\mathrm{d}\ell \\ $$$$\:=\mathrm{2}\sqrt{{x}}\:+\mathrm{ln}\:\mid\frac{\sqrt{\mathrm{x}}−\mathrm{1}}{\:\sqrt{{x}}\:+\mathrm{1}}\mid\:+\:\mathrm{C}\: \\ $$$$ \\ $$

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