integral of 1/(u^2-a^2)

Lösung

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

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

Schritte zur Lösung

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

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

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

Setze in

v = \frac{u}{a}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1}{3 ( 1 + 2 x ) ^{2}} d x

\int_{- 3}^{3} - 9 - x^{2} d x

\int (x - 3)^{- 1} d x

\int \frac{1 + x ^{8}}{x ^{13}} d x

t an g e n t 1 - 6 x, at x = - 1