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Beliebt Rechnen >

integral from 0 to 1 of 2/(x^2-4)

Lösung

\int_{0}^{1} \frac{2}{x ^{2} - 4} d x

Lösung

- \frac{ln ( 3 )}{2}

+1

Dezimale

- 0.54930 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{2}{x ^{2} - 4} d x

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{1} \frac{1}{x ^{2} - 4} d x

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

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

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

Entferne die Konstante:

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

= 2 \cdot \frac{1}{2} (- \int_{0}^{\frac{1}{2}} \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}

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{ln ( 2 ) + ln ( \frac{3}{2} )}{2}

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

Vereinfache

= - \frac{ln ( 3 )}{2}

Graph

Beliebte Beispiele

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\int_{\frac{π}{2}}^{π} sin (θ) d θ

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\int_{0}^{\infty} e^{- x} \cdot x d x

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\int_{0}^{1} (x^{2} - 1) - (- x + 2) d x

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