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Beliebt Rechnen >

integral from 1 to 4 of (x^2+7)/(5x-x^2)

Lösung

\int_{1}^{4} \frac{x ^{2} + 7}{5 x - x ^{2}} d x

Lösung

- 3 + \frac{78}{5} ln (2)

+1

Dezimale

7.81309 \dots

Schritte zur Lösung

\int_{1}^{4} \frac{x ^{2} + 7}{5 x - x ^{2}} d x

Faktorisiere

\frac{x ^{2} + 7}{5 x - x ^{2}}

= \int_{1}^{4} \frac{x ^{2} + 7}{- x ( x - 5 )} d x

Entferne die Konstante:

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

= - \int_{1}^{4} \frac{x ^{2} + 7}{x ( x - 5 )} d x

Multipliziere aus

\frac{x ^{2} + 7}{x ( x - 5 )} : \frac{x ^{2}}{x ( x - 5 )} + \frac{7}{x ( x - 5 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int_{1}^{4} \frac{x ^{2}}{x ( x - 5 )} d x + \int_{1}^{4} \frac{7}{x ( x - 5 )} d x)

\int_{1}^{4} \frac{x ^{2}}{x ( x - 5 )} d x = 3 - 10 ln (2)

\int_{1}^{4} \frac{7}{x ( x - 5 )} d x = - \frac{28}{5} ln (2)

= - (3 - 10 ln (2) - \frac{28}{5} ln (2))

Vereinfache

- (3 - 10 ln (2) - \frac{28}{5} ln (2)) : - 3 + \frac{78}{5} ln (2)

= - 3 + \frac{78}{5} ln (2)

Graph

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