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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int \frac{1}{u} + \frac{1}{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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - \frac{1}{u}

Setze in

u = x - 1

ein

= ln ∣ x - 1 ∣ - \frac{1}{x - 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

Graph

Beliebte Beispiele

integral from 0 to 3 of 2^x

\int_{0}^{3} 2^{x} d x

sum from n=0 to infinity of (n^3)/(2^n)

n = 0 \sum \infty \frac{n ^{3}}{2 ^{n}}

integral of 1/(sin^2(a)x)

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

y^'=x^5+2x^4-x-2

y^{^{'}} = x^{5} + 2 x^{4} - x - 2

laplacetransform t^{1/2}

l a pl a ce t r an s f or m t^{\frac{1}{2}}