What is the antiderivative of $xln(x) - x$ ?

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Answer 1 · 2016-06-28T08:26:31

$= x^2 / 2 ln x - 3/4x^2 + C$

$int dx \ (x ln x - x)$

for first term, $x ln x$ , we use IBP

$u = ln x, u' = 1/x$
$v' = x, v = x^2/2$

$\implies x^2/2 ln x - int dx \ (x/2)$

$= x^2 / 2 ln x - x^2 /4$

for the second term

$- int dx \ (x) = - x^2 / 2$

So the integral is

$= x^2 / 2 ln x - x^2 /4 color{red}{- x^2/2 + C}$

$= x^2 / 2 ln x - 3/4x^2 + C$

What is the antiderivative of xln(x) - xxln(x) - x?