What is the antiderivative of $\frac{ln x}{x^{\frac{1}{2}}}$ $(lnx) / (x^(1/2))$ ?

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Answer 1 · 2016-06-27T08:42:44

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

$int \ \ d(ln(x)) / (x^(1/2))x$

we can try IBP

Wikipedia

$u = ln (x), u' = 1/x$
$v' = x^{-1/2}, v = 2x^{1/2}$

so we have

$2x^{1/2} ln (x) - int \ 2x^{1/2}*1/x \ dx$

$= 2x^{1/2} ln (x) - 2 int \ x^{-1/2} \ dx$

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

$= 2sqrt{x} (ln (x) - 2 ) + C$

What is the antiderivative of (lnx) / (x^(1/2))lnxx12(lnx) / (x^(1/2))?