integral of x(x+1)^{10}

Lösung

\int x (x + 1)^{10} d x

\frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11} + C

Schritte zur Lösung

\int x (x + 1)^{10} d x

Wende U-Substitution an

= \int (u - 1) u^{10} d u

Multipliziere aus

(u - 1) u^{10} : u^{11} - u^{10}

= \int u^{11} - u^{10} 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

= \int u^{11} d u - \int u^{10} d u

\int u^{11} d u = \frac{u ^{12}}{12}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{12}}{12} - \frac{u ^{11}}{11}

Setze in

u = x + 1

ein

= \frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11}

Füge eine Konstante zur Lösung hinzu

= \frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11} + C

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