Integration by Parts Math Example 5

Follow the full solution, then compare it with the other examples linked below.

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Find

\displaystyle\int \ln x\,dx

1
Write as $\int \ln x \cdot 1\,dx$ . Let $u = \ln x$ , $dv = dx$ ; $du = \frac{dx}{x}$ , $v = x$ .
2
$x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C$ .

Answer

x\ln x - x + C

Writing

\ln x = \ln x \cdot 1

sets up IBP with

dv = dx

. This is the standard trick for integrating isolated logarithms.

About Integration by Parts

An integration technique based on the product rule: $\int u\,dv = uv - \int v\,du$ . Used when the integrand is a product of two functions.

Find [formula].

Find [formula].

Evaluate [formula] using integration by parts twice.

Find [formula].