∫log _ex dx =∫log _ex· 1dx =log _e x∫ 1dx ∫[d/dxlog _ex∫ 1 dx] dx =xlog _ex ∫1/x· xdx =xlog _ex ∫ 1 dx =x log _ex x+c , where ...

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Integration by Parts

Evaluate: ∫...

Question

Evaluate: $\int {log}_{e} x d x$ .

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Solution

$\int {log}_{e} x d x$
$= \int {log}_{e} x \cdot 1 d x$
$= {log}_{e} x \int 1 d x - \int [\frac{d}{d x} {log}_{e} x \int 1 d x] d x$
$= x {log}_{e} x - \int \frac{1}{x} \cdot x d x$
$= x {log}_{e} x - \int 1 d x$
$= x {log}_{e} x - x + c$ , where $C$ is integration constant.

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