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Beliebt Rechnen >

integral of s/(s^2+6s+11)

Lösung

\int \frac{s}{s ^{2} + 6 s + 11} d s

Lösung

\frac{1}{2} ln s^{2} + 6 s + 11 - \frac{3}{2} arctan (\frac{1}{2} (s + 3)) + C

Schritte zur Lösung

\int \frac{s}{s ^{2} + 6 s + 11} d s

Vervollständige das Quadrat

s^{2} + 6 s + 11 : (s + 3)^{2} + 2

= \int \frac{s}{( s + 3 ) ^{2} + 2} d s

Wende U-Substitution an

= \int \frac{u - 3}{u ^{2} + 2} d u

Multipliziere aus

\frac{u - 3}{u ^{2} + 2} : \frac{u}{u ^{2} + 2} - \frac{3}{u ^{2} + 2}

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} + 2} d u - \int \frac{3}{u ^{2} + 2} d u

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

\int \frac{3}{u ^{2} + 2} d u = \frac{3}{2} arctan (\frac{u}{2})

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

Setze in

u = s + 3

ein

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

Vereinfache

\frac{1}{2} ln (s + 3)^{2} + 2 - \frac{3}{2} arctan (\frac{s + 3}{2}) : \frac{1}{2} ln s^{2} + 6 s + 11 - \frac{3}{2} arctan (\frac{1}{2} (s + 3))

= \frac{1}{2} ln s^{2} + 6 s + 11 - \frac{3}{2} arctan (\frac{1}{2} (s + 3))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln s^{2} + 6 s + 11 - \frac{3}{2} arctan (\frac{1}{2} (s + 3)) + C

Graph

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