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

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Answer 1 · 2018-06-27T10:49:00

$int lnx/x^(1/2) dx = 2 sqrtx(lnx - 2)+C$

Integrate by parts:

$int lnx/x^(1/2) dx = int lnx * x^(-1/2)dx$

$int lnx/x^(1/2) dx = 2 int lnx * d/dx (x^(1/2)) dx$

$int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*d/dx(lnx)dx$

$int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*1/xdx$

$int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(-1/2)dx$

$int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 4 x^(+1/2)+C$

$int lnx/x^(1/2) dx = 2 x^(1/2)(lnx - 2)+C$

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