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Beliebt Rechnen >

integral of (2-u)/(u^2+2u+3)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{2}{u ^{2} + 2 u + 3} d u = 2 arctan (\frac{1}{2} (u + 1))

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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