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Beliebt Rechnen >

integral of (x+5)/(x^2-6x+13)

Lösung

\int \frac{x + 5}{x ^{2} - 6 x + 13} d x

Lösung

\frac{1}{2} ln x^{2} - 6 x + 13 + 4 arctan (\frac{1}{2} (x - 3)) + C

Schritte zur Lösung

\int \frac{x + 5}{x ^{2} - 6 x + 13} d x

Vervollständige das Quadrat

x^{2} - 6 x + 13 : (x - 3)^{2} + 4

= \int \frac{x + 5}{( x - 3 ) ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{u + 8}{u ^{2} + 4} d u

Multipliziere aus

\frac{u + 8}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} + \frac{8}{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{8}{u ^{2} + 4} d u

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

\int \frac{8}{u ^{2} + 4} d u = 4 arctan (\frac{u}{2})

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

Setze in

u = x - 3

ein

= \frac{1}{2} ln (x - 3)^{2} + 4 + 4 arctan (\frac{x - 3}{2})

Vereinfache

\frac{1}{2} ln (x - 3)^{2} + 4 + 4 arctan (\frac{x - 3}{2}) : \frac{1}{2} ln x^{2} - 6 x + 13 + 4 arctan (\frac{1}{2} (x - 3))

= \frac{1}{2} ln x^{2} - 6 x + 13 + 4 arctan (\frac{1}{2} (x - 3))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} - 6 x + 13 + 4 arctan (\frac{1}{2} (x - 3)) + C

Graph

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