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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x/((1-x^3))

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

integral of (y/((x+y)^2)-1)

\int (\frac{y}{( x + y ) ^{2}} - 1) d x

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

\int \frac{e ^{x}}{( e ^{2 x} + 1 )} d x

integral of sin(22x)

\int sin (22 x) d x

integral of 2/(t^3)sin(1/(t^2)-4)

\int \frac{2}{t ^{3}} sin (\frac{1}{t ^{2}} - 4) d t