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Beliebt Rechnen >

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

Lösung

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

Lösung

x^{2} - 2 x + ln x^{2} - 2 x + x - 1 + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} - 2 x : (x - 1)^{2} - 1

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

= u^{2} - 1 + ln u^{2} - 1 + u

Setze in

u = x - 1

ein

= (x - 1)^{2} - 1 + ln (x - 1)^{2} - 1 + x - 1

Vereinfache

(x - 1)^{2} - 1 + ln (x - 1)^{2} - 1 + x - 1 : x^{2} - 2 x + ln x^{2} - 2 x + x - 1

= x^{2} - 2 x + ln x^{2} - 2 x + x - 1

Füge eine Konstante zur Lösung hinzu

= x^{2} - 2 x + ln x^{2} - 2 x + x - 1 + C

Graph

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