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Beliebt Rechnen >

integral of 1/(x-2-sqrt(x))

Lösung

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

Lösung

2 (\frac{1}{3} ln x + 1 + \frac{2}{3} ln x - 2) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{u}{u ^{2} - 2 - u} : \frac{1}{3 ( u + 1 )} + \frac{2}{3 ( u - 2 )}

= 2 \cdot \int \frac{1}{3 ( u + 1 )} + \frac{2}{3 ( u - 2 )} 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}{3 ( u + 1 )} d u + \int \frac{2}{3 ( u - 2 )} d u)

\int \frac{1}{3 ( u + 1 )} d u = \frac{1}{3} ln ∣ u + 1 ∣

\int \frac{2}{3 ( u - 2 )} d u = \frac{2}{3} ln ∣ u - 2 ∣

= 2 (\frac{1}{3} ln ∣ u + 1 ∣ + \frac{2}{3} ln ∣ u - 2 ∣)

Setze in

u = x

ein

= 2 (\frac{1}{3} ln x + 1 + \frac{2}{3} ln x - 2)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{3} ln x + 1 + \frac{2}{3} ln x - 2) + C

Graph

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\int 1 cos (n x) d x

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\int \frac{1}{9 sec ( x )} d x

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