integral from 1 to e of (ln(x))/(x^p)

Lösung

\int_{1}^{e} \frac{ln ( x )}{x ^{p}} d x

\frac{- p e ^{- p + 1} + 1}{( 1 - p ) ^{2}}

Schritte zur Lösung

\int_{1}^{e} \frac{ln ( x )}{x ^{p}} d x

Wende die partielle Integration an

= [\frac{1}{- p + 1} (x^{- p + 1} ln (x) - \int \frac{1}{x ^{p} ( - p + 1 )} d x (- p + 1))]_{1}^{e}

\int \frac{1}{x ^{p} ( - p + 1 )} d x = \frac{x ^{- p + 1}}{( - p + 1 ) ^{2}}

= [\frac{1}{- p + 1} (x^{- p + 1} ln (x) - \frac{x ^{- p + 1}}{( - p + 1 ) ^{2}} (- p + 1))]_{1}^{e}

\frac{x ^{- p + 1}}{( - p + 1 ) ^{2}} (- p + 1) = \frac{x ^{- p + 1}}{- p + 1}

= [\frac{1}{- p + 1} (x^{- p + 1} ln (x) - \frac{x ^{- p + 1}}{- p + 1})]_{1}^{e}

Berechne die Grenzen:

- \frac{p e ^{- p + 1}}{( - p + 1 ) ^{2}} + \frac{1}{( - p + 1 ) ^{2}}

= - \frac{p e ^{- p + 1}}{( - p + 1 ) ^{2}} + \frac{1}{( - p + 1 ) ^{2}}

Vereinfache

= \frac{- p e ^{- p + 1} + 1}{( 1 - p ) ^{2}}

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