What is $\int \frac{ln x}{\sqrt{x}} d x$ $int ln x / sqrtx dx$ ?

Question

1664 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-03T23:32:13

$2 \sqrt{x} (ln x - 2) + c$ $2sqrtx(lnx-2)+c$

$\int \frac{ln x}{\sqrt{x}} d x = \int \frac{ln x}{2 \sqrt{x}} (2 d x)$ $intlnx/sqrtxdx=intlnx/color(blue)(2sqrtx)(2color(blue)(dx))$

Substitute $\sqrt{x} = u$ $sqrtx=u$

$=$ $=$ $intlnu^color(red)(2)/cancel(u)(2cancel(u)du)$ $=$

$int2*2lnudu$ $=$

$4intlnudu$ $=$

$4(u$ $lnu$ $-u)+c$ $=$

$4(sqrtxlnsqrtx-sqrtx)+c$ $=$

$4(1/2sqrtxlnx-sqrtx)+c$ $=$

$2sqrtx(lnx-2)+c$ ,

$c$ $in$ $RR$

What is int ln x / sqrtx dx∫lnx√xdxint ln x / sqrtx dx?