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Question Number 73545 by Rio Michael last updated on 13/Nov/19

$$evaluate\int lnxdx$$

Commented by Tony Lin last updated on 13/Nov/19

$$integrationbypart$$$$\int f{}^{\prime }\left(x\right)g\left(x\right)=f\left(x\right)g\left(x\right)-\int f\left(x\right)g{}^{\prime }\left(x\right)$$$$\int lnxdx$$$$=\int \left(1\times lnx\right)dx$$$$=xlnx-\int \left(x\times \frac{1}{x}\right)dx$$$$=xlnx-x+c$$

Commented by Rio Michael last updated on 13/Nov/19

$$thanks$$

Answered by ajfour last updated on 13/Nov/19

$$\frac{d}{dx}\left(x\ln x\right)=\ln x+1$$$$\Rightarrow d\left(x\ln x\right)=\left(\ln x\right)dx+dx$$$$Integrating$$$$\int d\left(x\ln x\right)=\int \left(\ln x\right)dx+\int dx$$$$x\ln x+c=\int \left(\ln x\right)dx+x$$$$\Rightarrow \int \ln xdx=x\ln x-x+c.$$$$$$

Commented by Rio Michael last updated on 13/Nov/19

$$iappreciatesir$$

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