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Beliebt Rechnen >

integral of 1/(a-bx^2)

Lösung

\int \frac{1}{a - b x ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{1}{a - b x ^{2}} d x

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

\int \frac{1}{1 - u ^{2}} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

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

Setze in

u = \frac{b}{a} x

ein

= \frac{1}{a b} \frac{ln \frac{b}{a} x + 1}{2} - \frac{ln \frac{b}{a} x - 1}{2}

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int \frac{2 x ^{2} - 5 x - 7}{x - 3} d x