How do you integrate $\int \frac{ln x}{x}$ $intlnx/x$ using substitution?

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Answer 1 · 2016-12-06T21:35:50

$\int \frac{ln x}{x} d x = \frac{1}{2} {ln}^{2} x + a$ $intlnx/xdx = 1/2ln^2x+a$

Let $u = ln x$ $u = lnx$ then $\frac{d u}{d x} = \frac{1}{x}$ $(du)/dx=1/x$ so we can write $d u = \frac{1}{x} d x$ $du=1/xdx$

Substituting into the integral we get:

$\ \ \ \ \ intlnx/xdx = int(lnx)(1/xdx)$
$:. intlnx/xdx = int(u)(du)$
$:. intlnx/xdx = intudu$
$:. intlnx/xdx = 1/2u^2+a$

And replacing $u$ we get:

$\ \ \ \ \ intlnx/xdx = 1/2ln^2x+a$

How do you integrate intlnx/x∫lnxxintlnx/x using substitution?