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

Lösung

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

\frac{x ^{6}}{6} + \frac{20}{11} x^{\frac{11}{2}} + 9 x^{5} + \frac{80}{3} x^{\frac{9}{2}} + \frac{105 x ^{4}}{2} + 72 x^{\frac{7}{2}} + 70 x^{3} + 48 x^{\frac{5}{2}} + \frac{45 x ^{2}}{2} + \frac{20}{3} x^{\frac{3}{2}} + x + C

Schritte zur Lösung

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

Multipliziere aus

(x + 1)^{10} : x^{5} + 10 x^{\frac{9}{2}} + 45 x^{4} + 120 x^{\frac{7}{2}} + 210 x^{3} + 252 x^{\frac{5}{2}} + 210 x^{2} + 120 x^{\frac{3}{2}} + 45 x + 10 x + 1

= \int x^{5} + 10 x^{\frac{9}{2}} + 45 x^{4} + 120 x^{\frac{7}{2}} + 210 x^{3} + 252 x^{\frac{5}{2}} + 210 x^{2} + 120 x^{\frac{3}{2}} + 45 x + 10 x + 1 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 x^{5} d x + \int 10 x^{\frac{9}{2}} d x + \int 45 x^{4} d x + \int 120 x^{\frac{7}{2}} d x + \int 210 x^{3} d x + \int 252 x^{\frac{5}{2}} d x + \int 210 x^{2} d x + \int 120 x^{\frac{3}{2}} d x + \int 45 x d x + \int 10 x d x + \int 1 d x

\int x^{5} d x = \frac{x ^{6}}{6}

\int 10 x^{\frac{9}{2}} d x = \frac{20}{11} x^{\frac{11}{2}}

\int 45 x^{4} d x = 9 x^{5}

\int 120 x^{\frac{7}{2}} d x = \frac{80}{3} x^{\frac{9}{2}}

\int 210 x^{3} d x = \frac{105 x ^{4}}{2}

\int 252 x^{\frac{5}{2}} d x = 72 x^{\frac{7}{2}}

\int 210 x^{2} d x = 70 x^{3}

\int 120 x^{\frac{3}{2}} d x = 48 x^{\frac{5}{2}}

\int 45 x d x = \frac{45 x ^{2}}{2}

\int 10 x d x = \frac{20}{3} x^{\frac{3}{2}}

\int 1 d x = x

= \frac{x ^{6}}{6} + \frac{20}{11} x^{\frac{11}{2}} + 9 x^{5} + \frac{80}{3} x^{\frac{9}{2}} + \frac{105 x ^{4}}{2} + 72 x^{\frac{7}{2}} + 70 x^{3} + 48 x^{\frac{5}{2}} + \frac{45 x ^{2}}{2} + \frac{20}{3} x^{\frac{3}{2}} + x

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{6}}{6} + \frac{20}{11} x^{\frac{11}{2}} + 9 x^{5} + \frac{80}{3} x^{\frac{9}{2}} + \frac{105 x ^{4}}{2} + 72 x^{\frac{7}{2}} + 70 x^{3} + 48 x^{\frac{5}{2}} + \frac{45 x ^{2}}{2} + \frac{20}{3} x^{\frac{3}{2}} + x + C