integral of 1/(1000x-x^2)

Lösung

\int \frac{1}{1000 x - x ^{2}} d x

\frac{1}{1000} (ln \frac{x - 500}{500} + 1 - ln \frac{x - 500}{500} - 1) + C

Schritte zur Lösung

\int \frac{1}{1000 x - x ^{2}} d x

Vervollständige das Quadrat

1000 x - x^{2} : - (x - 500)^{2} + 250000

= \int \frac{1}{- ( x - 500 ) ^{2} + 250000} d x

Wende U-Substitution an

= \int \frac{1}{- u ^{2} + 250000} d u

Wende integrale Substitution an

= \int \frac{1}{500 ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{500} \cdot \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}{500} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Ersetze zurück

= \frac{1}{500} (\frac{ln \frac{x - 500}{500} + 1}{2} - \frac{ln \frac{x - 500}{500} - 1}{2})

Vereinfache

\frac{1}{500} (\frac{ln \frac{x - 500}{500} + 1}{2} - \frac{ln \frac{x - 500}{500} - 1}{2}) : \frac{1}{1000} (ln \frac{x - 500}{500} + 1 - ln \frac{x - 500}{500} - 1)

= \frac{1}{1000} (ln \frac{x - 500}{500} + 1 - ln \frac{x - 500}{500} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{1000} (ln \frac{x - 500}{500} + 1 - ln \frac{x - 500}{500} - 1) + C

\int cos (2 x - 4) d x

\int \frac{1 - tan ^{2} ( x )}{sin ^{2} ( x )} d x

\int (6 x^{2} - 5) e^{2 x^{3} - 5 x} d x

\int (5 x + 16)^{4} d x

\int sin (3 x) cos^{- 4} (3 x) d x