integral from x to 1 of y^2-2y+1

Lösung

\int_{x}^{1} y^{2} - 2 y + 1 d y

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

Schritte zur Lösung

\int_{x}^{1} y^{2} - 2 y + 1 d y

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} y^{2} d y - \int_{x}^{1} 2 y d y + \int_{x}^{1} 1 d y

\int_{x}^{1} y^{2} d y = \frac{1 - x ^{3}}{3}

\int_{x}^{1} 2 y d y = 1 - x^{2}

\int_{x}^{1} 1 d y = 1 - x

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

Vereinfache

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

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