How do we integrate $\ln(x) $? I would like to know the steps, because I know the final answer, but confusing how to get it there

Question

How do I take the integral of $\ln(x)$ I would like to know the steps, because I know the final answer, but confusing how to get it there
$\int\ln(x) dx $

is it integration by parts ?

Answer 1

Yes, you can use integration by parts. Set $u = \ln(x)$, $dv = dx$, then $v = x$ and $du = \frac{1}{x} dx$, so

$$\int \ln(x) dx = \ln(x) x - \int x \frac{1}{x} dx = x \ln(x) - \int dx = x \ln(x) - x + C$$

Answer 2

Yes, you need to use this method. Just set $u=\ln x$ and $dv=dx$ in the following formula: $$\int u dv=uv-\int vdu$$

Answer 3

$\int u d(v) = uv - \int v d(u) $ is the integration by parts formula.

$$ \int \ln(x) dx = x\cdot \ln(x) - \int x d( \ln(x)) = x\cdot \ln(x) - \int x\frac 1x dx = x\cdot \ln(x) - x + C$$

Answer 4

You can substitute to get a familiar integral: let $x=e^t$, $dx=e^t dt$ and get

$$\int dt\: t\, e^t = t\, e^t - e^t +C = x \log{x}-x+C$$