integral from 1 to 4 of (5ln(x))/(x^2)

Lösung

\int_{1}^{4} \frac{5 ln ( x )}{x ^{2}} d x

5 (\frac{- 2 ln ( 2 ) - 1}{4} + 1)

Dezimale

2.01713 \dots

Schritte zur Lösung

\int_{1}^{4} \frac{5 ln ( x )}{x ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int_{1}^{4} \frac{ln ( x )}{x ^{2}} d x

Wende die partielle Integration an

= 5 [- \frac{ln ( x )}{x} - \int - \frac{1}{x ^{2}} d x]_{1}^{4}

\int - \frac{1}{x ^{2}} d x = \frac{1}{x}

= 5 [- \frac{ln ( x )}{x} - \frac{1}{x}]_{1}^{4}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= 5 [\frac{- ln ( x ) - 1}{x}]_{1}^{4}

Berechne die Grenzen:

\frac{- 2 ln ( 2 ) - 1}{4} + 1

= 5 (\frac{- 2 ln ( 2 ) - 1}{4} + 1)

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