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Question Number 146215 by mathdanisur last updated on 12/Jul/21

$$\int \frac{dx}{x\sqrt{x^2-4}}=?$$

Answered by Olaf_Thorendsen last updated on 12/Jul/21

$$\mathrm{F}\left(x\right)=\int \frac{dx}{x\sqrt{x^2-4}}$$$$\mathrm{F}\left(2\mathrm{ch}u\right)=\int \frac{2\mathrm{sh}udu}{2\mathrm{ch}u\sqrt{{4\mathrm{ch}}^{2}u-4}}$$$$\mathrm{F}\left(2\mathrm{ch}u\right)=\int \frac{\mathrm{sh}udu}{\mathrm{ch}u\times 2\mathrm{sh}u}$$$$\mathrm{F}\left(2\mathrm{ch}u\right)=\frac{1}{2}\int \frac{du}{\mathrm{ch}u}$$$$\mathrm{F}\left(2\mathrm{ch}u\right)=\int \frac{e^{u}du}{e^{2u}+1}$$$$\mathrm{F}\left(2\mathrm{ch}u\right)=\arctan \left(e^u\right)+\mathrm{C}$$$$\mathrm{F}\left(x\right)=\arctan \left(e^{\mathrm{argch}\frac{x}{2}}\right)+\mathrm{C}$$$$\mathrm{F}\left(x\right)=\arctan \left(e^{\ln (\frac{x}{2}+\sqrt{\frac{x^2}{4}-1})}\right)+\mathrm{C}$$$$\mathrm{F}\left(x\right)=\arctan (\frac{x}{2}+\sqrt{\frac{x^2}{4}-1})+\mathrm{C}$$

Answered by qaz last updated on 12/Jul/21

$$\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^2-4}}=\int \frac{\mathrm{dx}}{{\mathrm{x}}^2\sqrt{1-\frac{4}{{\mathrm{x}}^2}}}=-\frac{1}{2}\int \frac{\mathrm{d}\left(\frac{2}{\mathrm{x}}\right)}{\sqrt{1-\frac{4}{{\mathrm{x}}^2}}}=-\frac{1}{2}{\sin }^{-1}\frac{2}{\mathrm{x}}+\mathrm{C}$$

Answered by mathmax by abdo last updated on 12/Jul/21

$$\mathrm{\Phi }=\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^2-4}}\mathrm{we}\mathrm{do}\mathrm{the}\mathrm{changement}\mathrm{x}=\frac{2}{\mathrm{sint}}\Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=-\frac{2\mathrm{cost}}{{\sin }^2\mathrm{t}}$$$$\Rightarrow \mathrm{\Phi }=-2\int \frac{\mathrm{cost}}{{\sin }^2\mathrm{t}\times \frac{2}{\mathrm{sint}}\sqrt{\frac{4}{{\sin }^2\mathrm{t}}-4}}\mathrm{dt}$$$$=-\int \frac{\mathrm{cost}}{\mathrm{sint}\times 2\times \frac{\mathrm{cost}}{\mathrm{sint}}}\mathrm{dt}=-\frac{1}{2}\int \mathrm{dt}=-\frac{\mathrm{t}}{2}\mathrm{C}$$$$\mathrm{sint}=\frac{2}{\mathrm{x}}\Rightarrow \mathrm{t}=\arcsin \left(\frac{2}{\mathrm{x}}\right)\Rightarrow \mathrm{\Phi }=-\frac{1}{2}\arcsin \left(\frac{2}{\mathrm{x}}\right)+\mathrm{C}$$$$$$

Commented by mathdanisur last updated on 12/Jul/21

$$ThanksSer$$

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