Problems on Techniques of Integration

Use the Integration by Parts for definite integrals. Set

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

Then

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

So

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

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

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

Detailed Answer.

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