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Beliebt Rechnen >

integral from 1 to x^2 of 2x^2y(x+y)

Lösung

\int_{1}^{x^{2}} 2 x^{2} y (x + y) d y

Lösung

x^{7} - x^{3} + 2 x^{2} \frac{x ^{6} - 1}{3}

Schritte zur Lösung

\int_{1}^{x^{2}} 2 x^{2} y (x + y) d y

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 x^{2} \cdot \int_{1}^{x^{2}} (x + y) y d y

Multipliziere aus

(x + y) y : x y + y^{2}

= 2 x^{2} \cdot \int_{1}^{x^{2}} x y + y^{2} 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

= 2 x^{2} (\int_{1}^{x^{2}} x y d y + \int_{1}^{x^{2}} y^{2} d y)

\int_{1}^{x^{2}} x y d y = \frac{x ( x ^{4} - 1 )}{2}

\int_{1}^{x^{2}} y^{2} d y = \frac{x ^{6} - 1}{3}

= 2 x^{2} (\frac{x ( x ^{4} - 1 )}{2} + \frac{x ^{6} - 1}{3})

Vereinfache

2 x^{2} (\frac{x ( x ^{4} - 1 )}{2} + \frac{x ^{6} - 1}{3}) : x^{7} - x^{3} + 2 x^{2} \frac{x ^{6} - 1}{3}

= x^{7} - x^{3} + 2 x^{2} \frac{x ^{6} - 1}{3}

Beliebte Beispiele

integral from 1 to e^8 of 4/x

\int_{1}^{e^{8}} \frac{4}{x} d x

integral from-1 to 2 of (2x+1)(2x-1)

\int_{- 1}^{2} (2 x + 1) (2 x - 1) d x

integral from 0 to 1 of 8x^7(6-x^8)

\int_{0}^{1} 8 x^{7} (6 - x^{8}) d x

integral from-3 to 4 of (1-x)

\int_{- 3}^{4} (1 - x) d x

integral from 0 to 1 of e^y-1

\int_{0}^{1} e^{y} - 1 d y