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Beliebt Rechnen >

integral of x/(x^2-2x+5)

Lösung

\int \frac{x}{x ^{2} - 2 x + 5} d x

Lösung

\frac{1}{2} ln x^{2} - 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} - 2 x + 5} d x

Vervollständige das Quadrat

x^{2} - 2 x + 5 : (x - 1)^{2} + 4

= \int \frac{x}{( x - 1 ) ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{u + 1}{u ^{2} + 4} d u

Multipliziere aus

\frac{u + 1}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} + \frac{1}{u ^{2} + 4}

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} + 4} d u + \int \frac{1}{u ^{2} + 4} d u

\int \frac{u}{u ^{2} + 4} d u = \frac{1}{2} ln u^{2} + 4

\int \frac{1}{u ^{2} + 4} d u = \frac{1}{2} arctan (\frac{u}{2})

= \frac{1}{2} ln u^{2} + 4 + \frac{1}{2} arctan (\frac{u}{2})

Setze in

u = x - 1

ein

= \frac{1}{2} ln (x - 1)^{2} + 4 + \frac{1}{2} arctan (\frac{x - 1}{2})

Vereinfache

\frac{1}{2} ln (x - 1)^{2} + 4 + \frac{1}{2} arctan (\frac{x - 1}{2}) : \frac{1}{2} ln x^{2} - 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x - 1))

= \frac{1}{2} ln x^{2} - 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x - 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} - 2 x + 5 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) + C

Graph

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