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

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Answer 1 · 2017-01-09T13:29:06

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

We can calculate the integral by parts:

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

Solving this last integral:

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

Putting it together:

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

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