Expert answers to "Evaluate the following integral. ∫x2⋅ln⁡xdx"

College
Calculus and Analysis
Integral Calculus
Applications of integrals

Nann

Answered question

2021-10-14

Evaluate the following integral.

\int x^{2} \cdot \ln x d x

Answer & Explanation

wheezym

Skilled2021-10-15Added 103 answers

Step 1
To assess the integrals, given
$\int x^{2} \ln x d x$
formula: part-by-part integration
$\int u d v = u v - \int v d u$
Step 2
To assess the integrals, given
$\int x^{2} \ln x d x$
Put $u = \ln x \Rightarrow d u = \frac{d x}{x}$
and $d v = x^{2} d x \Rightarrow v = \frac{x^{3}}{3}$
Therefore, $\int x^{2} \ln x d x = \frac{x^{3}}{3} \ln x - \int \frac{x^{3}}{3} \frac{d x}{x}$
$= \frac{x^{3}}{3} \ln x - \frac{x^{3}}{9} + C$
$= \frac{x^{3}}{3} (\ln x - \frac{1}{3}) + C$
$= \frac{x^{3}}{9} (3 \ln x - \ln e) + C$
$= \frac{x^{3}}{9} \ln (\frac{x^{3}}{e}) + C$
Therefore, $\int x^{2} \ln x d x = \frac{x^{3}}{9} \ln (\frac{x^{3}}{e}) + C$

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