What is $\int \frac{ln (ln x)}{x} d x$ $int ln(lnx)/xdx$ ?

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Answer 1 · 2015-11-03T19:38:49

Consider the following substitution.

Let:
$t = ln x$ $t = lnx$
$d t = \frac{1}{x} d x$ $dt = 1/xdx$

$\int \frac{ln (ln x)}{x} d x = \int ln t d t$ $intln(lnx)/(x)dx = intlntdt$

$\int u d v = u v - \int v d u$ $int udv = uv - intvdu$

Let:
$u = ln t$ $u = lnt$
$d u = \frac{1}{t} d t$ $du = 1/tdt$
$d v = d t$ $dv = dt$
$v = t$ $v = t$

$= ln t - \int t \cdot \frac{1}{t} d t$ $=lnt - int t*1/tdt$

$= t ln t - t$ $= tlnt - t$

And now let's substitute back in our original variables.

$= (ln x) ln (ln x) - ln x + C$ $= color(blue)((lnx)ln(lnx) - lnx + C)$

What is ∫ln(lnx)xdxint ln(lnx)/xdx?