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Beliebt Rechnen >

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

Lösung

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

Lösung

x - 2 ln ∣ x ∣ + 2 arctan (x + 1) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\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)

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

Vereinfache

= x - 2 ln ∣ x ∣ + 2 arctan (x + 1)

Füge eine Konstante zur Lösung hinzu

= x - 2 ln ∣ x ∣ + 2 arctan (x + 1) + C

Graph

Beliebte Beispiele

integral of sin^2(5x)*cos^3(5x)

\int sin^{2} (5 x) \cdot cos^{3} (5 x) d x

integral of x(sqrt(3x+4))

\int x (3 x + 4) d x

integral of e^{(1x)/2}

\int e^{\frac{1 x}{2}} d x

integral of x^2csch^2(x^3)

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

integral of 2xln(x+1)

\int 2 x ln (x + 1) d x