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Beliebt Rechnen >

integral of (sqrt(x-2))/(x-3)

Lösung

\int \frac{x - 2}{x - 3} d x

Lösung

- ln x - 2 + 1 + ln x - 2 - 1 + 2 x - 2 + C

Schritte zur Lösung

\int \frac{x - 2}{x - 3} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

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

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

Entferne die Konstante:

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

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

\frac{u ^{2}}{- u ^{2} + 1} = \frac{1}{- u ^{2} + 1} - 1

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d u = u

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

Setze in

u = x - 2

ein

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

Vereinfache

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

= - ln x - 2 + 1 + ln x - 2 - 1 + 2 x - 2

Füge eine Konstante zur Lösung hinzu

= - ln x - 2 + 1 + ln x - 2 - 1 + 2 x - 2 + C

Graph

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