How do you integrate $\frac{ln x}{x^{\frac{1}{2}}}$ $ln x / x^(1/2)$ ?

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Answer 1 · 2016-07-05T17:00:06

$= 2 \sqrt{x} ln x - 4 \sqrt{x} + C$ $= 2 sqrt x ln x - 4 sqrt x + C$

Use IBP

$int u v' = uv - int u'v$

here

$u = ln x, u' = 1/x$
$v' = x^{-1/2}, v = 2 x^{1/2}$ , using the power rule

so we have

$2 sqrt x ln x - 2 int dx qquad 1/x sqrt x$

$= 2 sqrt x ln x - 2 int dx qquad x^{-1/2}$

$= 2 sqrt x ln x - 2*x^{1/2}/(1/2) + C$

$= 2 sqrt x ln x - 4 sqrt x + C$

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