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Beliebt Rechnen >

integral of (x-3)/(x^2+4x+8)

Lösung

\int \frac{x - 3}{x ^{2} + 4 x + 8} d x

Lösung

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

Schritte zur Lösung

\int \frac{x - 3}{x ^{2} + 4 x + 8} d x

Multipliziere aus

\frac{x - 3}{x ^{2} + 4 x + 8} : \frac{x}{x ^{2} + 4 x + 8} - \frac{3}{x ^{2} + 4 x + 8}

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}{x ^{2} + 4 x + 8} d x - \int \frac{3}{x ^{2} + 4 x + 8} d x

\int \frac{x}{x ^{2} + 4 x + 8} d x = \frac{1}{2} ln \frac{1}{4} (x + 2)^{2} + 1 - arctan (\frac{1}{2} (x + 2))

\int \frac{3}{x ^{2} + 4 x + 8} d x = \frac{3}{2} arctan (\frac{1}{2} (x + 2))

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

Vereinfache

\frac{1}{2} ln \frac{1}{4} (x + 2)^{2} + 1 - arctan (\frac{1}{2} (x + 2)) - \frac{3}{2} arctan (\frac{1}{2} (x + 2)) : \frac{1}{2} ln \frac{1}{4} x^{2} + x + 2 - \frac{5}{2} arctan (\frac{1}{2} x + 1)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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