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

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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