Problems on Techniques of Integration

The logarithmic function is one of these functions whose derivative is very nice (rational function). So it is a good idea to use the integration by parts technique in which $\ln(x)$ will be differentiated. One may argue that we do not have two functions to apply this technique. That's true except that $\ln(x)$ may also be seen as the product of the functions $1$ and $\ln(x)$.

Set

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

Then

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

Since

$$\int u dv = u v - \int v du\;,$$

we get

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

which implies

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

or

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

It is a common mistake to forget the constant $C$.

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