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Beliebt Rechnen >

integral of (x+1)/(2x^2+4x+3)

Lösung

\int \frac{x + 1}{2 x ^{2} + 4 x + 3} d x

Lösung

\frac{1}{4} ln 2 x^{2} + 4 x + 3 + C

Schritte zur Lösung

\int \frac{x + 1}{2 x ^{2} + 4 x + 3} d x

Multipliziere aus

\frac{x + 1}{2 x ^{2} + 4 x + 3} : \frac{x}{2 x ^{2} + 4 x + 3} + \frac{1}{2 x ^{2} + 4 x + 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{x}{2 x ^{2} + 4 x + 3} d x + \int \frac{1}{2 x ^{2} + 4 x + 3} d x

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

\int \frac{1}{2 x ^{2} + 4 x + 3} d x = \frac{1}{2} arctan (2 (x + 1))

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

Addiere gleiche Elemente:

- \frac{1}{2} arctan (2 (x + 1)) + \frac{1}{2} arctan (2 (x + 1)) = 0

= \frac{1}{4} ln 2 x^{2} + 4 x + 3

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln 2 x^{2} + 4 x + 3 + C

Graph

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