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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln x^{2} + 2 x + 2 - 6 arctan (x + 1) + 2 ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of e^{xln(2)}

\int e^{x l n (2)} d x

integral of (2x^2-x)/x

\int \frac{2 x ^{2} - x}{x} d x

integral of (sin^3(θ)+1)/(sin^2(θ))

\int \frac{sin ^{3} ( θ ) + 1}{sin ^{2} ( θ )} d θ

integral of 1/(3x+9)

\int \frac{1}{3 x + 9} d x

integral of (x^5-5^x+\sqrt[5]{x}-5/x)

\int (x^{5} - 5^{x} + 5 x - \frac{5}{x}) d x