What's the integral of ln(x) dx

June 19, 2026Mathematics

ScanSolve AI Answer

Answer

x\ln(x) - x + C

4 steps

Okay, let's integrate $\ln(x)$ . This requires integration by parts.

Set up integration by parts.
We use the formula $\int u\,dv = uv - \int v\,du$ .
Let $u = \ln(x)$ and $dv = dx$ .
Find $du$ and $v$ .
Differentiate $u$ to find $du$ :
$du = \frac{1}{x}\,dx$
Integrate $dv$ to find $v$ :
$v = \int dx = x$
Apply the integration by parts formula.
Substitute $u$ , $v$ , $du$ , and $dv$ into the formula:
$\int \ln(x)\,dx = (\ln(x))(x) - \int (x)\left(\frac{1}{x}\right)\,dx$
Simplify and integrate the remaining term.
$\int \ln(x)\,dx = x\ln(x) - \int 1\,dx$ $\int \ln(x)\,dx = x\ln(x) - x + C$
The integral of $\ln(x)\,dx$ is $\boxed{x\ln(x) - x + C}$ .