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

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Answer 1 · 2016-06-22T14:34:04

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

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

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

$\implies 2 sqrt{x} ln(2x) - int (2 sqrt(x))/x \ dx$

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

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

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

What is the antiderivative of ln(2x)/x^(1/2)ln(2x)x12ln(2x)/x^(1/2)?