integral of (2x+1)/(x^2+2x+5)

Lösung

\int \frac{2 x + 1}{x ^{2} + 2 x + 5} d x

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

Schritte zur Lösung

\int \frac{2 x + 1}{x ^{2} + 2 x + 5} d x

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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