integral of x^{1/3}(x^{4/3}+9)^8

解

\int x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} d x

\frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}} + C

解答ステップ

\int x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} d x

拡張

x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} : x^{11} + 72 x^{\frac{29}{3}} + 2268 x^{\frac{25}{3}} + 40824 x^{7} + 459270 x^{\frac{17}{3}} + 3306744 x^{\frac{13}{3}} + 14880348 x^{3} + 38263752 x^{\frac{5}{3}} + 43046721 x^{\frac{1}{3}}

= \int x^{11} + 72 x^{\frac{29}{3}} + 2268 x^{\frac{25}{3}} + 40824 x^{7} + 459270 x^{\frac{17}{3}} + 3306744 x^{\frac{13}{3}} + 14880348 x^{3} + 38263752 x^{\frac{5}{3}} + 43046721 x^{\frac{1}{3}} 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^{11} d x + \int 72 x^{\frac{29}{3}} d x + \int 2268 x^{\frac{25}{3}} d x + \int 40824 x^{7} d x + \int 459270 x^{\frac{17}{3}} d x + \int 3306744 x^{\frac{13}{3}} d x + \int 14880348 x^{3} d x + \int 38263752 x^{\frac{5}{3}} d x + \int 43046721 x^{\frac{1}{3}} d x

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

\int 72 x^{\frac{29}{3}} d x = \frac{27}{4} x^{\frac{32}{3}}

\int 2268 x^{\frac{25}{3}} d x = 243 x^{\frac{28}{3}}

\int 40824 x^{7} d x = 5103 x^{8}

\int 459270 x^{\frac{17}{3}} d x = \frac{137781}{2} x^{\frac{20}{3}}

\int 3306744 x^{\frac{13}{3}} d x = \frac{1240029}{2} x^{\frac{16}{3}}

\int 14880348 x^{3} d x = 3720087 x^{4}

\int 38263752 x^{\frac{5}{3}} d x = 14348907 x^{\frac{8}{3}}

\int 43046721 x^{\frac{1}{3}} d x = \frac{129140163}{4} x^{\frac{4}{3}}

= \frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}}

定数を解答に追加する

= \frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}} + C