∫ x log x dx =

Concept:

The product of function can be integrated by the method of "Integration by parts".

By the method of integration by parts we have,

\(\smallint f\left( x \right)g\left( x \right)dx\; = \;f\left( x \right)\smallint g\left( x \right)dx - \smallint \left[ {f'\left( x \right)\smallint g\left( x \right)dx} \right]dx\)

Where f is the first function and g is the second function.

Which is chosen based on the order for the selection of the first function: ILATE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)

Calculation:

Given:

We have to simplify the expression ∫ x log x dx

Here nothing has been mentioned so, we will take the base of the log as e.

And as we know, from integration by parts,

log x = u and x = v

which leads to,

\(I = \log x \times \frac{{{x^2}}}{2} - \frac{1}{2}\smallint {x^2} \times \frac{1}{x}dx\)

\(I = \log x \times \frac{{{x^2}}}{2} - \frac{1}{2} \times \frac{{{x^2}}}{2}\)

\(I=\frac{{{x^2}}}{2}\left( {\log x - \frac{1}{2}} \right)\)