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Beliebt Rechnen >

integral from 0 to 1 of (80)/(x^2-100)

Lösung

\int_{0}^{1} \frac{80}{x ^{2} - 100} d x

Lösung

- 4 (ln (\frac{11}{10}) - ln (\frac{9}{10}))

+1

Dezimale

- 0.80268 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{80}{x ^{2} - 100} d x

Entferne die Konstante:

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

= 80 \cdot \int_{0}^{1} \frac{1}{x ^{2} - 100} d x

Wende integrale Substitution an

= 80 \cdot \int_{0}^{\frac{1}{10}} \frac{1}{10 ( u ^{2} - 1 )} d u

Entferne die Konstante:

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

= 80 \cdot \frac{1}{10} \cdot \int_{0}^{\frac{1}{10}} \frac{1}{u ^{2} - 1} d u

Faktorisiere

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

= 80 \cdot \frac{1}{10} \cdot \int_{0}^{\frac{1}{10}} \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 80 \cdot \frac{1}{10} (- \int_{0}^{\frac{1}{10}} \frac{1}{- u ^{2} + 1} d u)

Nutze das gemeinsame Integral :

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

= 80 \cdot \frac{1}{10} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{10}})

Vereinfache

80 \cdot \frac{1}{10} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{10}}) : - 8 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{10}}

= - 8 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{10}}

Berechne die Grenzen:

\frac{ln ( \frac{11}{10} ) - ln ( \frac{9}{10} )}{2}

= - 8 \cdot \frac{ln ( \frac{11}{10} ) - ln ( \frac{9}{10} )}{2}

Vereinfache

= - 4 (ln (\frac{11}{10}) - ln (\frac{9}{10}))

Graph

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