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

Lösung

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

y^{2} - 8 y + 16

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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

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

\int_{- 2 + y}^{2} 2 d x = - 2 y + 8

= 2 (- (4 y - y^{2} - 2 + \frac{( - 2 + y ) ^{2}}{2}) - 2 y + 8)

Vereinfache

2 (- (4 y - y^{2} - 2 + \frac{( - 2 + y ) ^{2}}{2}) - 2 y + 8) : y^{2} - 8 y + 16

= y^{2} - 8 y + 16

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