integral of x^2(x^3+5)^{10}

解

\int x^{2} (x^{3} + 5)^{10} d x

\frac{x ^{33}}{33} + \frac{5 x ^{30}}{3} + \frac{125 x ^{27}}{3} + 625 x^{24} + 6250 x^{21} + 43750 x^{18} + 218750 x^{15} + 781250 x^{12} + 1953125 x^{9} + \frac{9765625 x ^{6}}{3} + \frac{9765625 x ^{3}}{3} + C

解答ステップ

\int x^{2} (x^{3} + 5)^{10} d x

拡張

x^{2} (x^{3} + 5)^{10} : x^{32} + 50 x^{29} + 1125 x^{26} + 15000 x^{23} + 131250 x^{20} + 787500 x^{17} + 3281250 x^{14} + 9375000 x^{11} + 17578125 x^{8} + 19531250 x^{5} + 9765625 x^{2}

= \int x^{32} + 50 x^{29} + 1125 x^{26} + 15000 x^{23} + 131250 x^{20} + 787500 x^{17} + 3281250 x^{14} + 9375000 x^{11} + 17578125 x^{8} + 19531250 x^{5} + 9765625 x^{2} d x

総和規則を適用する:

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

= \int x^{32} d x + \int 50 x^{29} d x + \int 1125 x^{26} d x + \int 15000 x^{23} d x + \int 131250 x^{20} d x + \int 787500 x^{17} d x + \int 3281250 x^{14} d x + \int 9375000 x^{11} d x + \int 17578125 x^{8} d x + \int 19531250 x^{5} d x + \int 9765625 x^{2} d x

\int x^{32} d x = \frac{x ^{33}}{33}

\int 50 x^{29} d x = \frac{5 x ^{30}}{3}

\int 1125 x^{26} d x = \frac{125 x ^{27}}{3}

\int 15000 x^{23} d x = 625 x^{24}

\int 131250 x^{20} d x = 6250 x^{21}

\int 787500 x^{17} d x = 43750 x^{18}

\int 3281250 x^{14} d x = 218750 x^{15}

\int 9375000 x^{11} d x = 781250 x^{12}

\int 17578125 x^{8} d x = 1953125 x^{9}

\int 19531250 x^{5} d x = \frac{9765625 x ^{6}}{3}

\int 9765625 x^{2} d x = \frac{9765625 x ^{3}}{3}

= \frac{x ^{33}}{33} + \frac{5 x ^{30}}{3} + \frac{125 x ^{27}}{3} + 625 x^{24} + 6250 x^{21} + 43750 x^{18} + 218750 x^{15} + 781250 x^{12} + 1953125 x^{9} + \frac{9765625 x ^{6}}{3} + \frac{9765625 x ^{3}}{3}

定数を解答に追加する

= \frac{x ^{33}}{33} + \frac{5 x ^{30}}{3} + \frac{125 x ^{27}}{3} + 625 x^{24} + 6250 x^{21} + 43750 x^{18} + 218750 x^{15} + 781250 x^{12} + 1953125 x^{9} + \frac{9765625 x ^{6}}{3} + \frac{9765625 x ^{3}}{3} + C