integral from-3 to x^2 of (x-5)(x^2+1)

Lösung

\int_{- 3}^{x^{2}} (x - 5) (x^{2} + 1) d x

\frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

Schritte zur Lösung

\int_{- 3}^{x^{2}} (x - 5) (x^{2} + 1) d x

Multipliziere aus

(x - 5) (x^{2} + 1) : x^{3} + x - 5 x^{2} - 5

= \int_{- 3}^{x^{2}} x^{3} + x - 5 x^{2} - 5 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_{- 3}^{x^{2}} x^{3} d x + \int_{- 3}^{x^{2}} x d x - \int_{- 3}^{x^{2}} 5 x^{2} d x - \int_{- 3}^{x^{2}} 5 d x

\int_{- 3}^{x^{2}} x^{3} d x = \frac{x ^{8} - 81}{4}

\int_{- 3}^{x^{2}} x d x = \frac{x ^{4} - 9}{2}

\int_{- 3}^{x^{2}} 5 x^{2} d x = 5 (\frac{x ^{6}}{3} + 9)

\int_{- 3}^{x^{2}} 5 d x = 5 x^{2} + 15

= \frac{x ^{8} - 81}{4} + \frac{x ^{4} - 9}{2} - 5 (\frac{x ^{6}}{3} + 9) - (5 x^{2} + 15)

Vereinfache

\frac{x ^{8} - 81}{4} + \frac{x ^{4} - 9}{2} - 5 (\frac{x ^{6}}{3} + 9) - (5 x^{2} + 15) : \frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

= \frac{1}{4} (x^{8} - 81) - \frac{5 x ^{6}}{3} + \frac{1}{2} (x^{4} - 9) - 5 x^{2} - 60

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