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Question Number 94948 by rb222 last updated on 22/May/20

$${\int }_0^{\sqrt{2ln\left(3\right)}}{xe}^{\frac{x^2}{2}}dx=...$$

Commented by Tony Lin last updated on 22/May/20

$$let\frac{x^2}{2}=u,du=xdx$$$${\int }_0^{ln3}{e}^{u}du$$$$=3-1=2$$

Commented by rb222 last updated on 22/May/20

$$thankssir$$

Commented by john santu last updated on 22/May/20

$$\int x.e^{\frac{x^2}{2}}dx=\int e^{\frac{1}{2}{x}^2}d\left(\frac{1}{2}{x}^2\right)$$$$=e^{\frac{1}{2}{x}^2}$$$$then\underset{0}{\stackrel{\sqrt{2\ln \left(3\right)}}{\int }}x.e^{\frac{1}{2}{x}^2}dx={\left[e^{\frac{1}{2}{x}^2}\right]}_0^{\sqrt{2\ln \left(3\right)}}$$$$=e^{\ln \left(3\right)}-{\mathrm{e}}^0=3-1=2$$

Answered by niroj last updated on 22/May/20

$${\int }_0^{\sqrt{2In\left(3\right)}}\mathrm{x}{\mathrm{e}}^{\frac{{\mathrm{x}}^2}{2}}\mathrm{dx}$$$$\mathrm{Put},\frac{{\mathrm{x}}^2}{2}=\mathrm{t}$$$${\mathrm{x}}^2=2\mathrm{t}$$$$2\mathrm{xdx}=2\mathrm{dt}$$$$\mathrm{xdx}=\mathrm{dt}$$$$\mathrm{If}\mathrm{x}=\sqrt{2\ln \left(3\right)}\Rightarrow \mathrm{t}=\frac{2\mathrm{In}\left(3\right)}{2}=\mathrm{In}\left(3\right)$$$$\mathrm{If}\mathrm{x}=0\Rightarrow \mathrm{t}=0$$$${\int }_0^{\mathrm{In}\left(3\right)}{\mathrm{e}}^{\mathrm{t}}\mathrm{dt}$$$$={\left[{\mathrm{e}}^{\mathrm{t}}\right]}_0^{\ln \left(3\right)}$$$$={\mathrm{e}}^{\ln \left(3\right)}-{\mathrm{e}}^0$$$$=3-1=2//.$$$$$$

Commented by rb222 last updated on 22/May/20

$$thankssir$$

Commented by niroj last updated on 22/May/20

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Commented by peter frank last updated on 22/May/20

$$\mathrm{thank}\mathrm{you}$$

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