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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

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\int 9 + e^{2 x} - \frac{5}{x ^{3}} d x

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integral of (2x^2-3x)

\int (2 x^{2} - 3 x) d x