integral of 1/(u^2-2)

Lösung

\int \frac{1}{u ^{2} - 2} d u

- \frac{1}{2 2} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1) + C

Schritte zur Lösung

\int \frac{1}{u ^{2} - 2} d u

Wende integrale Substitution an

= \int \frac{1}{2 ( v ^{2} - 1 )} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} - 1} d v

Faktorisiere

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

= \frac{1}{2} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{2} (- \int \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}

= \frac{1}{2} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Setze in

v = \frac{u}{2}

ein

= \frac{1}{2} - \frac{ln \frac{u}{2} + 1}{2} - \frac{ln \frac{u}{2} - 1}{2}

Vereinfache

\frac{1}{2} - \frac{ln \frac{u}{2} + 1}{2} - \frac{ln \frac{u}{2} - 1}{2} : - \frac{1}{2 2} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1)

= - \frac{1}{2 2} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 2} (ln \frac{u}{2} + 1 - ln \frac{u}{2} - 1) + C

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