Problems on Techniques of Integration

Problem. Evaluate

$$\int x^n \ln(x) dx\;\;\;\mbox{where}\;\;n = 1,2,\cdots$$

Short Answer. Use the Integration by Parts. Set

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

Then

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

So

$$\int x^n \ln(x) dx = \frac{x^{n+1}}{n+1} \ln(x) - \int \frac{... ...{1}{x} dx = \frac{x^{n+1}}{n+1} \ln(x) - \int \frac{x^n}{n+1}dx$$

or

$$\int x^n \ln(x) dx = \frac{x^{n+1}}{n+1} \ln(x) - \frac{x^{n+1}}{(n+1)^2} + C\;.$$

Detailed Answer.

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