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Beliebt Rechnen >

integral from 0 to 1 of (x-8)/(x^2-5x+6)

Lösung

\int_{0}^{1} \frac{x - 8}{x ^{2} - 5 x + 6} d x

Lösung

\frac{- ln ( 3 ) + 11 ln ( \frac{3}{4} )}{2}

+1

Dezimale

- 2.13155 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{x - 8}{x ^{2} - 5 x + 6} d x

Wende U-Substitution an

= \int_{- 8}^{- 7} \frac{u}{u ^{2} + 11 u + 30} d u

Vervollständige das Quadrat

u^{2} + 11 u + 30 : (u + \frac{11}{2})^{2} - \frac{1}{4}

= \int_{- 8}^{- 7} \frac{u}{( u + \frac{11}{2} ) ^{2} - \frac{1}{4}} d u

Wende U-Substitution an

= \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{2 ( 2 v - 11 )}{4 v ^{2} - 1} d v

Entferne die Konstante:

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

= 2 \cdot \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{2 v - 11}{4 v ^{2} - 1} d v

Multipliziere aus

\frac{2 v - 11}{4 v ^{2} - 1} : \frac{2 v}{4 v ^{2} - 1} - \frac{11}{4 v ^{2} - 1}

Wende die Summenregel an:

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

= 2 (\int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{2 v}{4 v ^{2} - 1} d v - \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{11}{4 v ^{2} - 1} d v)

\int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{2 v}{4 v ^{2} - 1} d v = - \frac{1}{4} ln (3)

\int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{11}{4 v ^{2} - 1} d v = - \frac{11 ln ( \frac{3}{4} )}{4}

= 2 (- \frac{1}{4} ln (3) - (- \frac{11 ln ( \frac{3}{4} )}{4}))

Vereinfache

2 (- \frac{1}{4} ln (3) - (- \frac{11 ln ( \frac{3}{4} )}{4})) : \frac{- ln ( 3 ) + 11 ln ( \frac{3}{4} )}{2}

= \frac{- ln ( 3 ) + 11 ln ( \frac{3}{4} )}{2}

Graph

Beliebte Beispiele

derivative of (x^2/(ln(x)))

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\frac{d}{d x} (x^{\frac{3}{5}} x^{3} + 2 x - 6)

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\frac{d}{d x} (2 x^{2} - 9 x + 10)

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