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Question Number 144975 by mim24 last updated on 01/Jul/21

Answered by ArielVyny last updated on 01/Jul/21

$$\int u^{\prime }\times u^{n}du=\frac{1}{n+1}{u}^{n+1}+cte$$$$\int (1+\frac{1}{x})(x+ln\left(x\right))dx=\frac{1}{2}{(x+lnx)}^2$$

Answered by som(math1967) last updated on 01/Jul/21

$$let(x+lnx)=z$$$$(1+\frac{1}{x})dx=dz$$$$\int \mathit{z}\mathit{d}\mathit{z}=\frac{{\mathit{z}}^2}{2}+\mathit{C}=\frac{{(\mathit{x}+\mathit{l}\mathit{n}\mathit{x})}^2}{2}+\mathit{C}$$

Answered by imjagoll last updated on 01/Jul/21

$$\mathrm{let}\mathrm{u}=\mathrm{x}+\ln \mathrm{x}\Rightarrow \mathrm{du}=(1+\frac{1}{\mathrm{x}})\mathrm{dx}$$$$\mathrm{I}=\int \mathrm{u}\mathrm{du}=\frac{{\mathrm{u}}^2}{2}+\mathrm{c}=\frac{{(\mathrm{x}+\ln \mathrm{x})}^2}{2}+\mathrm{c}$$$$=\frac{{\left(\ln {\mathrm{xe}}^{\mathrm{x}}\right)}^2}{2}+\mathrm{c}$$

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