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Beliebt Rechnen >

integral of 1/(5-4x^2)

Lösung

\int \frac{1}{5 - 4 x ^{2}} d x

Lösung

\frac{1}{4 5} (ln \frac{2}{5} x + 1 - ln \frac{2}{5} x - 1) + C

Schritte zur Lösung

\int \frac{1}{5 - 4 x ^{2}} d x

Wende integrale Substitution an

= \int \frac{1}{2 5 ( 1 - u ^{2} )} d u

Entferne die Konstante:

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

= \frac{1}{2 5} \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}{2 5} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Setze in

u = \frac{2}{5} x

ein

= \frac{1}{2 5} \frac{ln \frac{2}{5} x + 1}{2} - \frac{ln \frac{2}{5} x - 1}{2}

Vereinfache

\frac{1}{2 5} \frac{ln \frac{2}{5} x + 1}{2} - \frac{ln \frac{2}{5} x - 1}{2} : \frac{1}{4 5} (ln \frac{2}{5} x + 1 - ln \frac{2}{5} x - 1)

= \frac{1}{4 5} (ln \frac{2}{5} x + 1 - ln \frac{2}{5} x - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4 5} (ln \frac{2}{5} x + 1 - ln \frac{2}{5} x - 1) + C

Graph

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