integral from 0 to 1/10 of x/(1+8x^3)

Lösung

\int_{0}^{\frac{1}{10}} \frac{x}{1 + 8 x ^{3}} d x

\frac{1}{4} - \frac{1}{3} ln (\frac{6}{5}) - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3}

Dezimale

0.00498 \dots

Schritte zur Lösung

\int_{0}^{\frac{1}{10}} \frac{x}{1 + 8 x ^{3}} d x

Wende U-Substitution an

= \int_{1}^{\frac{126}{125}} \frac{1}{12 u 3 u - 1} d u

Entferne die Konstante:

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

= \frac{1}{12} \cdot \int_{1}^{\frac{126}{125}} \frac{1}{u 3 u - 1} d u

Wende U-Substitution an

= \frac{1}{12} \cdot \int_{0}^{\frac{1}{125}} \frac{1}{( v + 1 ) 3 v} d v

Wende U-Substitution an

= \frac{1}{12} \cdot \int_{0}^{\frac{1}{5}} \frac{3 w}{w ^{3} + 1} d w

Entferne die Konstante:

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

= \frac{1}{12} \cdot 3 \cdot \int_{0}^{\frac{1}{5}} \frac{w}{w ^{3} + 1} d w

Ermittle den Partialbruch von

\frac{w}{w ^{3} + 1} : - \frac{1}{3 ( w + 1 )} + \frac{w + 1}{3 ( w ^{2} - w + 1 )}

= \frac{1}{12} \cdot 3 \cdot \int_{0}^{\frac{1}{5}} - \frac{1}{3 ( w + 1 )} + \frac{w + 1}{3 ( w ^{2} - w + 1 )} d w

Wende die Summenregel an:

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

= \frac{1}{12} \cdot 3 (- \int_{0}^{\frac{1}{5}} \frac{1}{3 ( w + 1 )} d w + \int_{0}^{\frac{1}{5}} \frac{w + 1}{3 ( w ^{2} - w + 1 )} d w)

\int_{0}^{\frac{1}{5}} \frac{1}{3 ( w + 1 )} d w = \frac{1}{3} ln (\frac{6}{5})

\int_{0}^{\frac{1}{5}} \frac{w + 1}{3 ( w ^{2} - w + 1 )} d w = - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3}

= \frac{1}{12} \cdot 3 - \frac{1}{3} ln (\frac{6}{5}) - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3}

Vereinfache

\frac{1}{12} \cdot 3 - \frac{1}{3} ln (\frac{6}{5}) - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3} : \frac{1}{4} - \frac{1}{3} ln (\frac{6}{5}) - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3}

= \frac{1}{4} - \frac{1}{3} ln (\frac{6}{5}) - \frac{1}{6} ln (\frac{25}{21}) + \frac{- 6 arctan ( \frac{3}{5} ) + π}{6 3}

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