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Beliebt Rechnen >

integral from 0 to 1 of (12)/(x^2-4)

Lösung

\int_{0}^{1} \frac{12}{x ^{2} - 4} d x

Lösung

- 3 ln (3)

+1

Dezimale

- 3.29583 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{12}{x ^{2} - 4} d x

Entferne die Konstante:

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

= 12 \cdot \int_{0}^{1} \frac{1}{x ^{2} - 4} d x

Wende integrale Substitution an

= 12 \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

= 12 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{1}{2}} \frac{1}{u ^{2} - 1} d u

Faktorisiere

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

= 12 \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

= 12 \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}

= 12 \cdot \frac{1}{2} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}})

Vereinfache

12 \cdot \frac{1}{2} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}}) : - 6 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{2}}

= - 6 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{2}}

Berechne die Grenzen:

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

= - 6 \cdot \frac{ln ( 2 ) + ln ( \frac{3}{2} )}{2}

Vereinfache

= - 3 ln (3)

Graph

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