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Beliebt Rechnen >

integral of (x^2)/(sqrt(5-x))

Lösung

\int \frac{x ^{2}}{5 - x} d x

Lösung

- 2 (\frac{1}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + 25 5 - x) + C

Schritte zur Lösung

\int \frac{x ^{2}}{5 - x} d x

Wende U-Substitution an

= \int - 2 (- u^{2} + 5)^{2} d u

Entferne die Konstante:

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

= - 2 \cdot \int (- u^{2} + 5)^{2} d u

Multipliziere aus

(- u^{2} + 5)^{2} : u^{4} - 10 u^{2} + 25

= - 2 \cdot \int u^{4} - 10 u^{2} + 25 d u

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 u^{4} d u - \int 10 u^{2} d u + \int 25 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 10 u^{2} d u = \frac{10 u ^{3}}{3}

\int 25 d u = 25 u

= - 2 (\frac{u ^{5}}{5} - \frac{10 u ^{3}}{3} + 25 u)

Setze in

u = 5 - x

ein

= - 2 (\frac{( 5 - x ) ^{5}}{5} - \frac{10 ( 5 - x ) ^{3}}{3} + 25 5 - x)

Vereinfache

- 2 (\frac{( 5 - x ) ^{5}}{5} - \frac{10 ( 5 - x ) ^{3}}{3} + 25 5 - x) : - 2 (\frac{1}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + 25 5 - x)

= - 2 (\frac{1}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + 25 5 - x)

Füge eine Konstante zur Lösung hinzu

= - 2 (\frac{1}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + 25 5 - x) + C

Graph

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