integral of x(x+5)^{21}

Lösung

\int x (x + 5)^{21} d x

\frac{1}{506} (x + 5)^{22} (22 x - 5) + C

Schritte zur Lösung

\int x (x + 5)^{21} d x

Wende U-Substitution an

= \int \frac{21 u ( 21 u - 5 )}{21} d u

Entferne die Konstante:

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

= \frac{1}{21} \cdot \int 21 u (21 u - 5) d u

Multipliziere aus

21 u (21 u - 5) : u^{\frac{2}{21}} - 5 21 u

= \frac{1}{21} \cdot \int u^{\frac{2}{21}} - 5 21 u 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

= \frac{1}{21} (\int u^{\frac{2}{21}} d u - \int 5 21 u d u)

\int u^{\frac{2}{21}} d u = \frac{21}{23} u^{\frac{23}{21}}

\int 5 21 u d u = \frac{105}{22} u^{\frac{22}{21}}

= \frac{1}{21} (\frac{21}{23} u^{\frac{23}{21}} - \frac{105}{22} u^{\frac{22}{21}})

Setze in

u = (x + 5)^{21}

ein

= \frac{1}{21} (\frac{21}{23} ((x + 5)^{21})^{\frac{23}{21}} - \frac{105}{22} ((x + 5)^{21})^{\frac{22}{21}})

Vereinfache

\frac{1}{21} (\frac{21}{23} ((x + 5)^{21})^{\frac{23}{21}} - \frac{105}{22} ((x + 5)^{21})^{\frac{22}{21}}) : \frac{1}{506} (x + 5)^{22} (22 x - 5)

= \frac{1}{506} (x + 5)^{22} (22 x - 5)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{506} (x + 5)^{22} (22 x - 5) + C

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