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Beliebt Rechnen >

integral of ((x^3+4x^2+8x+5)/(x^2+2x+1))

Lösung

\int (\frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1}) d x

Lösung

\frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2} + C

Schritte zur Lösung

\int \frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1} d x

Multipliziere aus

\frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1} : \frac{x ^{3}}{x ^{2} + 2 x + 1} + \frac{4 x ^{2}}{x ^{2} + 2 x + 1} + \frac{8 x}{x ^{2} + 2 x + 1} + \frac{5}{x ^{2} + 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 ^{3}}{x ^{2} + 2 x + 1} d x + \int \frac{4 x ^{2}}{x ^{2} + 2 x + 1} d x + \int \frac{8 x}{x ^{2} + 2 x + 1} d x + \int \frac{5}{x ^{2} + 2 x + 1} d x

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

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

\int \frac{8 x}{x ^{2} + 2 x + 1} d x = 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1})

\int \frac{5}{x ^{2} + 2 x + 1} d x = - \frac{5}{x + 1}

= \frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1} + 4 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣)) + 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1}) - \frac{5}{x + 1}

Vereinfache

\frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1} + 4 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣)) + 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1}) - \frac{5}{x + 1} : \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2}

= \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2} + C

Graph

Beliebte Beispiele

integral of 1/(sqrt(x^2)-4)

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

integral of x(\sqrt[3]{x^2-1})

\int x (3 x^{2} - 1) d x

integral of 4e^{-1/2 x}

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

integral of (x+1)/(4x^2-x-5)

\int \frac{x + 1}{4 x ^{2} - x - 5} d x

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

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