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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x ^{2} + 2 x + 5} d x

Vervollständige das Quadrat

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

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

Wende U-Substitution an

= 2 \cdot \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

= 2 (\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})

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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