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#