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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x^{-2})/(sqrt((2-x)(2+x)))

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

integral of (10+2x^2+2y^2)

\int (10 + 2 x^{2} + 2 y^{2}) d x

integral of sqrt(v^2+2v)

\int v^{2} + 2 v d v

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

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

integral of x^22^{-x}

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