Problems on Techniques of Integration

Use the Integration by Parts. Set

$$\left\{\begin{array}{lll} u &=& \ln(x)\\ dv &=& x dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&\displaystyle \frac{1}{x} dx\\ &&\\ v &=& \displaystyle \frac{x^2}{2}\;. \end{array}\right.$$

So

$$\int x \ln(x) dx = \frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \frac{1}{x} dx = \frac{x^2}{2} \ln(x) - \int \frac{x}{2}dx$$

or

$$\int x \ln(x) dx = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C\;.$$

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