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Beliebt Rechnen >

integral of (x^2+4x)/(x^3-x^2+x-1)

Lösung

\int \frac{x ^{2} + 4 x}{x ^{3} - x ^{2} + x - 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} + 4 x}{x ^{3} - x ^{2} + x - 1} d x

Multipliziere aus

\frac{x ^{2} + 4 x}{x ^{3} - x ^{2} + x - 1} : \frac{x ^{2}}{x ^{3} - x ^{2} + x - 1} + \frac{4 x}{x ^{3} - x ^{2} + x - 1}

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 ^{2}}{x ^{3} - x ^{2} + x - 1} d x + \int \frac{4 x}{x ^{3} - x ^{2} + x - 1} d x

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

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

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

integral of sin(18x)cos(15x)

\int sin (18 x) cos (15 x) d x

integral of 1/(\sqrt[3]{x-9)}

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

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

\int (\frac{8 x ^{3} - 4 x ^{2} - 3 x}{6 x}) d x