What is the antiderivative of $ln(x)$ ?

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Answer 1 · 2015-03-05T21:18:53

$I = intln(x)dx$

Let $ln(x) = t$

$=> x=e^t$

$=> dx = e^tdt$

Substituting in the Integral,

$I = intte^tdt$

On integrating by parts, keeping the first function as $t$ and second function as $e^t$ , we get

$I = tinte^tdt - int(dt/dtinte^tdt)dt$

Which is, simply,

$I = te^t - e^t + C$

$=> I = e^t ( t-1) + C$

Substituting the value of $t = ln(x)$ ,

$intln(x)dx = x[ln(x) - 1] + C$

What is the antiderivative of ln(x)ln(x)?