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Question Number 84879 by sahnaz last updated on 17/Mar/20

$${\mathrm{e}}^{\int \frac{2\mathrm{dx}}{\mathrm{xlnx}}}$$

Commented by jagoll last updated on 17/Mar/20

$$\int \frac{2\mathrm{dx}}{\mathrm{x}\mathrm{lnx}}=\int \frac{2\mathrm{d}\left(\mathrm{lnx}\right)}{\mathrm{lnx}}=\int 2\frac{\mathrm{du}}{\mathrm{u}}$$$$=2\ln \mathrm{u}+\mathrm{c},[\mathrm{u}=\ln \mathrm{x}]$$$$=2\ln \left(\mathrm{lnx}\right)+2\mathrm{lnC}=2\ln \left(\mathrm{Clnx}\right)$$$${\mathrm{e}}^{\int \frac{2\mathrm{dx}}{\mathrm{lnx}}}={\mathrm{e}}^{\left[\ln {\left(\mathrm{Clnx}\right)}^2\right]}={\left(\mathrm{Cln}\mathrm{x}\right)}^2$$

Answered by bshahid010@gmail.com last updated on 17/Mar/20

$$\underset{\mathrm{e}}{\stackrel{2}{\int }}\frac{\mathrm{dx}}{\mathrm{x}.\mathrm{lnx}}\mathrm{put}\mathrm{lnx}=\mathrm{t}\Rightarrow \frac{1}{\mathrm{x}}\mathrm{dx}=\mathrm{dt}$$$$\text{Missing left or extra right}$$$$\ln \left(\ln \left(2\right)\right)$$$$$$$$$$

Commented by john santu last updated on 17/Mar/20

$$\mathrm{the}\mathrm{question}\mathrm{e}{}^{\int \frac{2\mathrm{dx}}{\mathrm{x}\ln \mathrm{x}}}\mathrm{sir}$$

Answered by bshahid010@gmail.com last updated on 18/Mar/20

$${\mathrm{e}}^{\int \frac{2\mathrm{dx}}{\mathrm{xlnx}}}\mathrm{put}\mathrm{lnx}=\mathrm{t}\Rightarrow \frac{1}{\mathrm{x}}\mathrm{dx}=\mathrm{dt}$$$${\mathrm{e}}^{2\int \frac{\mathrm{dt}}{\mathrm{t}}}={\mathrm{e}}^{2\ln \left(\mathrm{t}\right)+\mathrm{c}}$$$${\mathrm{e}}^{2\ln \left(\mathrm{lnx}\right)+\mathrm{c}}$$

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