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Beliebt Rechnen >

integral of (3x+6)/(x^2+2x+2)

Lösung

\int \frac{3 x + 6}{x ^{2} + 2 x + 2} d x

Lösung

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

Schritte zur Lösung

\int \frac{3 x + 6}{x ^{2} + 2 x + 2} d x

Multipliziere aus

\frac{3 x + 6}{x ^{2} + 2 x + 2} : \frac{3 x}{x ^{2} + 2 x + 2} + \frac{6}{x ^{2} + 2 x + 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{3 x}{x ^{2} + 2 x + 2} d x + \int \frac{6}{x ^{2} + 2 x + 2} d x

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

\int \frac{6}{x ^{2} + 2 x + 2} d x = 6 arctan (x + 1)

= 3 (\frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1)) + 6 arctan (x + 1)

Vereinfache

3 (\frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1)) + 6 arctan (x + 1) : \frac{3}{2} ln x^{2} + 2 x + 2 + 3 arctan (x + 1)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from 0 to 4 of sqrt(x)+3x

\int_{0}^{4} x + 3 x d x

integral of (x^6-x^3+64)/(x^4+8x^2)

\int \frac{x ^{6} - x ^{3} + 64}{x ^{4} + 8 x ^{2}} d x

derivative of 1-ln(1-x)

\frac{d}{d x} (1 - ln (1 - x))

derivative of 6xy-12

\frac{d}{d x} (6 x y - 12)

integral of 1/(2x-3)

\int \frac{1}{2 x - 3} d x