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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

x (x - 1 - (x - 1)^{2}) : x x - 1 - x^{3} + 2 x^{2} - x

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

\int x x - 1 d x = \frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{2}{3} (x - 1)^{\frac{3}{2}}

\int x^{3} d x = \frac{x ^{4}}{4}

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

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

Füge eine Konstante zur Lösung hinzu

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

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