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Beliebt >

integral from 1 to 2 of 1/(x^2-7x+3)

Lösung

\int_{1}^{2} \frac{1}{x ^{2} - 7 x + 3} d x

- \frac{ln ( \frac{79 + 11 37}{42} )}{37}

Dezimale Notation

- 0.20472 \dots

Schritte zur Lösung

\int_{1}^{2} \frac{1}{x ^{2} - 7 x + 3} d x

Vervollständige das Quadrat

x^{2} - 7 x + 3 : (x - \frac{7}{2})^{2} - \frac{37}{4}

= \int_{1}^{2} \frac{1}{( x - \frac{7}{2} ) ^{2} - \frac{37}{4}} d x

Wende U-Substitution an

: \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{4}{4 u ^{2} - 37} d u

= \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{4}{4 u ^{2} - 37} d u

Entferne die Konstante:

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

= 4 \cdot \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{1}{4 u ^{2} - 37} d u

Wende integrale Substitution an

= 4 \cdot \int_{- \frac{5}{37}}^{- \frac{3}{37}} \frac{1}{2 37 ( v ^{2} - 1 )} d v

Entferne die Konstante:

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

= 4 \cdot \frac{1}{2 37} \cdot \int_{- \frac{5}{37}}^{- \frac{3}{37}} \frac{1}{v ^{2} - 1} d v

Faktorisiere

v^{2} - 1 : - (- v^{2} + 1)

= 4 \cdot \frac{1}{2 37} \cdot \int_{- \frac{5}{37}}^{- \frac{3}{37}} \frac{1}{- ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= 4 \cdot \frac{1}{2 37} (- \int_{- \frac{5}{37}}^{- \frac{3}{37}} \frac{1}{- v ^{2} + 1} d v)

Nutze das gemeinsame Integral :

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= 4 \cdot \frac{1}{2 37} (- [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{- \frac{5}{37}}^{- \frac{3}{37}})

Vereinfache

4 \cdot \frac{1}{2 37} (- [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{- \frac{5}{37}}^{- \frac{3}{37}}) : - \frac{2}{37} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{- \frac{5}{37}}^{- \frac{3}{37}}

= - \frac{2}{37} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{- \frac{5}{37}}^{- \frac{3}{37}}

Berechne die Grenzen:

\frac{ln ( - \frac{3}{37} + 1 ) - ln ( \frac{3}{37} + 1 ) - ln ( - \frac{5}{37} + 1 ) + ln ( \frac{5}{37} + 1 )}{2}

= - \frac{2}{37} \cdot \frac{ln ( - \frac{3}{37} + 1 ) - ln ( \frac{3}{37} + 1 ) - ln ( - \frac{5}{37} + 1 ) + ln ( \frac{5}{37} + 1 )}{2}

Vereinfache

= - \frac{ln ( \frac{79 + 11 37}{42} )}{37}

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