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. There are basically two ways of doing this problem. You may try to find the antiderivative of $x^2 \ln(x)$ then takes its values at $e$ and $1$. Or you may want to use the integration by parts for definite integrals.

Set

$$\left\{\begin{array}{lll} u &=& \ln(x)\\ dv &=& x^2 dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&\displaystyle \frac{1}{x} dx\\ &&\\ v &=& \displaystyle \frac{x^3}{3}\;. \end{array}\right.$$

Since

$$\int_a^b u dv = [u v]_a^b - \int_a^b v du\;,$$

we get

$$\int_1^e x^2 \ln(x) dx = \left[\frac{x^3}{3} \ln(x)\right]_1^... ...left[\frac{x^3}{3} \ln(x)\right]_1^e - \int_1^e \frac{x^2}{3}dx$$

which implies

$$\int_1^e x^2 \ln(x) dx = \left[\frac{x^3}{3} \ln(x)\right]_1^... ..._1^e = \frac{e^3}{3} - \left(\frac{e^3}{9} - \frac{1}{9}\right)$$

or

$$\int_1^e x^2 \ln(x) dx = \frac{(2e^3 + 1)}{9}\cdot$$

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