integral of x/((x^2+2x+3))

Lösung

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

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

Schritte zur Lösung

\int \frac{x}{x ^{2} + 2 x + 3} d x

Vervollständige das Quadrat

x^{2} + 2 x + 3 : (x + 1)^{2} + 2

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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