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Beliebt Rechnen >

integral of x/(x^2+2x+4)

Lösung

\int \frac{x}{x ^{2} + 2 x + 4} d x

Lösung

\frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 2 x + 4} d x

Vervollständige das Quadrat

x^{2} + 2 x + 4 : (x + 1)^{2} + 3

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

Wende U-Substitution an

= \int \frac{u - 1}{u ^{2} + 3} d u

Multipliziere aus

\frac{u - 1}{u ^{2} + 3} : \frac{u}{u ^{2} + 3} - \frac{1}{u ^{2} + 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{u}{u ^{2} + 3} d u - \int \frac{1}{u ^{2} + 3} d u

\int \frac{u}{u ^{2} + 3} d u = \frac{1}{2} ln u^{2} + 3

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

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

Setze in

u = x + 1

ein

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

Vereinfache

\frac{1}{2} ln (x + 1)^{2} + 3 - \frac{1}{3} arctan (\frac{x + 1}{3}) : \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

= \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) + C

Graph

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