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Beliebt Rechnen >

integral of 1/(y^2-2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende integrale Substitution an

= \int \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

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

Faktorisiere

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

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

Entferne die Konstante:

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

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

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

Setze in

u = \frac{y}{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

laplacetransform 2(t-2)^2+6(t-2)+6

l a pl a ce t r an s f or m 2 (t - 2)^{2} + 6 (t - 2) + 6

integral of 1/(sqrt(4x^2-16))

\int \frac{1}{4 x ^{2} - 16} d x

integral of 2/(x^2-4x+13)

\int \frac{2}{x ^{2} - 4 x + 13} d x

sum from n=0 to infinity of 1/(n^4)

n = 0 \sum \infty \frac{1}{n ^{4}}

(dy)/(dx)+3x^2y=18x^2,y(0)=8

\frac{d y}{d x} + 3 x^{2} y = 18 x^{2}, y (0) = 8