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Beliebt Rechnen >

integral from 1 to 2 of (x+1)/(x(x^2+1))

Lösung

\int_{1}^{2} \frac{x + 1}{x ( x ^{2} + 1 )} d x

Lösung

- \frac{1}{2} ln (5) + \frac{3}{2} ln (2) - \frac{π}{4} + arctan (2)

+1

Dezimale

0.55675 \dots

Schritte zur Lösung

\int_{1}^{2} \frac{x + 1}{x ( x ^{2} + 1 )} d x

Ermittle den Partialbruch von

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

= \int_{1}^{2} \frac{1}{x} + \frac{- x + 1}{x ^{2} + 1} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{2} \frac{1}{x} d x + \int_{1}^{2} \frac{- x + 1}{x ^{2} + 1} d x

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

\int_{1}^{2} \frac{- x + 1}{x ^{2} + 1} d x = - \frac{1}{2} (ln (5) - ln (2)) + arctan (2) - \frac{π}{4}

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

Vereinfache

ln (2) - \frac{1}{2} (ln (5) - ln (2)) + arctan (2) - \frac{π}{4} : - \frac{1}{2} ln (5) + \frac{3}{2} ln (2) - \frac{π}{4} + arctan (2)

= - \frac{1}{2} ln (5) + \frac{3}{2} ln (2) - \frac{π}{4} + arctan (2)

Graph

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