integral of (x^5+4x^2)(x^3+1)^{12}

解答

\int (x^{5} + 4 x^{2}) (x^{3} + 1)^{12} d x

\frac{x ^{42}}{42} + \frac{16 x ^{39}}{39} + \frac{19 x ^{36}}{6} + \frac{44 x ^{33}}{3} + \frac{275 x ^{30}}{6} + \frac{308 x ^{27}}{3} + \frac{341 x ^{24}}{2} + \frac{1496 x ^{21}}{7} + \frac{407 x ^{18}}{2} + \frac{440 x ^{15}}{3} + \frac{473 x ^{12}}{6} + \frac{92 x ^{9}}{3} + \frac{49 x ^{6}}{6} + \frac{4 x ^{3}}{3} + C

求解步骤

\int (x^{5} + 4 x^{2}) (x^{3} + 1)^{12} d x

乘开

(x^{5} + 4 x^{2}) (x^{3} + 1)^{12} : x^{41} + 16 x^{38} + 114 x^{35} + 484 x^{32} + 1375 x^{29} + 2772 x^{26} + 4092 x^{23} + 4488 x^{20} + 3663 x^{17} + 2200 x^{14} + 946 x^{11} + 276 x^{8} + 49 x^{5} + 4 x^{2}

= \int x^{41} + 16 x^{38} + 114 x^{35} + 484 x^{32} + 1375 x^{29} + 2772 x^{26} + 4092 x^{23} + 4488 x^{20} + 3663 x^{17} + 2200 x^{14} + 946 x^{11} + 276 x^{8} + 49 x^{5} + 4 x^{2} d x

使用积分加法定则:

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

= \int x^{41} d x + \int 16 x^{38} d x + \int 114 x^{35} d x + \int 484 x^{32} d x + \int 1375 x^{29} d x + \int 2772 x^{26} d x + \int 4092 x^{23} d x + \int 4488 x^{20} d x + \int 3663 x^{17} d x + \int 2200 x^{14} d x + \int 946 x^{11} d x + \int 276 x^{8} d x + \int 49 x^{5} d x + \int 4 x^{2} d x

\int x^{41} d x = \frac{x ^{42}}{42}

\int 16 x^{38} d x = \frac{16 x ^{39}}{39}

\int 114 x^{35} d x = \frac{19 x ^{36}}{6}

\int 484 x^{32} d x = \frac{44 x ^{33}}{3}

\int 1375 x^{29} d x = \frac{275 x ^{30}}{6}

\int 2772 x^{26} d x = \frac{308 x ^{27}}{3}

\int 4092 x^{23} d x = \frac{341 x ^{24}}{2}

\int 4488 x^{20} d x = \frac{1496 x ^{21}}{7}

\int 3663 x^{17} d x = \frac{407 x ^{18}}{2}

\int 2200 x^{14} d x = \frac{440 x ^{15}}{3}

\int 946 x^{11} d x = \frac{473 x ^{12}}{6}

\int 276 x^{8} d x = \frac{92 x ^{9}}{3}

\int 49 x^{5} d x = \frac{49 x ^{6}}{6}

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

= \frac{x ^{42}}{42} + \frac{16 x ^{39}}{39} + \frac{19 x ^{36}}{6} + \frac{44 x ^{33}}{3} + \frac{275 x ^{30}}{6} + \frac{308 x ^{27}}{3} + \frac{341 x ^{24}}{2} + \frac{1496 x ^{21}}{7} + \frac{407 x ^{18}}{2} + \frac{440 x ^{15}}{3} + \frac{473 x ^{12}}{6} + \frac{92 x ^{9}}{3} + \frac{49 x ^{6}}{6} + \frac{4 x ^{3}}{3}

解答补常数

= \frac{x ^{42}}{42} + \frac{16 x ^{39}}{39} + \frac{19 x ^{36}}{6} + \frac{44 x ^{33}}{3} + \frac{275 x ^{30}}{6} + \frac{308 x ^{27}}{3} + \frac{341 x ^{24}}{2} + \frac{1496 x ^{21}}{7} + \frac{407 x ^{18}}{2} + \frac{440 x ^{15}}{3} + \frac{473 x ^{12}}{6} + \frac{92 x ^{9}}{3} + \frac{49 x ^{6}}{6} + \frac{4 x ^{3}}{3} + C

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