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

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Answer 1 · 2017-02-21T04:40:19

$\frac{{ln}^{2} (x)}{2} + C$ $ln^2(x)/2 +C$

$I = \int \frac{ln x}{x} d x$ $I=int lnx/x dx$

Let $u = ln x$ $u=lnx$

$\frac{d u}{d x} = \frac{1}{x} \to d x = x d u$ $(du)/dx = 1/x -> dx=x du$ [Standard differential]

$:. I= int u/x * x du = int u du$

$= u^2/2$ [Power rule]

Undo u substitution:

$I = ln^2 (x)/2 +C$

What is the antiderivative of ln(x)/xln(x)xln(x)/x?