integral of x^2(x^3+1)^{15}

解答

\int x^{2} (x^{3} + 1)^{15} d x

\frac{x ^{48}}{48} + \frac{x ^{45}}{3} + \frac{5 x ^{42}}{2} + \frac{35 x ^{39}}{3} + \frac{455 x ^{36}}{12} + 91 x^{33} + \frac{1001 x ^{30}}{6} + \frac{715 x ^{27}}{3} + \frac{2145 x ^{24}}{8} + \frac{715 x ^{21}}{3} + \frac{1001 x ^{18}}{6} + 91 x^{15} + \frac{455 x ^{12}}{12} + \frac{35 x ^{9}}{3} + \frac{5 x ^{6}}{2} + \frac{x ^{3}}{3} + C

求解步骤

\int x^{2} (x^{3} + 1)^{15} d x

乘开

x^{2} (x^{3} + 1)^{15} : x^{47} + 15 x^{44} + 105 x^{41} + 455 x^{38} + 1365 x^{35} + 3003 x^{32} + 5005 x^{29} + 6435 x^{26} + 6435 x^{23} + 5005 x^{20} + 3003 x^{17} + 1365 x^{14} + 455 x^{11} + 105 x^{8} + 15 x^{5} + x^{2}

= \int x^{47} + 15 x^{44} + 105 x^{41} + 455 x^{38} + 1365 x^{35} + 3003 x^{32} + 5005 x^{29} + 6435 x^{26} + 6435 x^{23} + 5005 x^{20} + 3003 x^{17} + 1365 x^{14} + 455 x^{11} + 105 x^{8} + 15 x^{5} + 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^{47} d x + \int 15 x^{44} d x + \int 105 x^{41} d x + \int 455 x^{38} d x + \int 1365 x^{35} d x + \int 3003 x^{32} d x + \int 5005 x^{29} d x + \int 6435 x^{26} d x + \int 6435 x^{23} d x + \int 5005 x^{20} d x + \int 3003 x^{17} d x + \int 1365 x^{14} d x + \int 455 x^{11} d x + \int 105 x^{8} d x + \int 15 x^{5} d x + \int x^{2} d x

\int x^{47} d x = \frac{x ^{48}}{48}

\int 15 x^{44} d x = \frac{x ^{45}}{3}

\int 105 x^{41} d x = \frac{5 x ^{42}}{2}

\int 455 x^{38} d x = \frac{35 x ^{39}}{3}

\int 1365 x^{35} d x = \frac{455 x ^{36}}{12}

\int 3003 x^{32} d x = 91 x^{33}

\int 5005 x^{29} d x = \frac{1001 x ^{30}}{6}

\int 6435 x^{26} d x = \frac{715 x ^{27}}{3}

\int 6435 x^{23} d x = \frac{2145 x ^{24}}{8}

\int 5005 x^{20} d x = \frac{715 x ^{21}}{3}

\int 3003 x^{17} d x = \frac{1001 x ^{18}}{6}

\int 1365 x^{14} d x = 91 x^{15}

\int 455 x^{11} d x = \frac{455 x ^{12}}{12}

\int 105 x^{8} d x = \frac{35 x ^{9}}{3}

\int 15 x^{5} d x = \frac{5 x ^{6}}{2}

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

= \frac{x ^{48}}{48} + \frac{x ^{45}}{3} + \frac{5 x ^{42}}{2} + \frac{35 x ^{39}}{3} + \frac{455 x ^{36}}{12} + 91 x^{33} + \frac{1001 x ^{30}}{6} + \frac{715 x ^{27}}{3} + \frac{2145 x ^{24}}{8} + \frac{715 x ^{21}}{3} + \frac{1001 x ^{18}}{6} + 91 x^{15} + \frac{455 x ^{12}}{12} + \frac{35 x ^{9}}{3} + \frac{5 x ^{6}}{2} + \frac{x ^{3}}{3}

解答补常数

= \frac{x ^{48}}{48} + \frac{x ^{45}}{3} + \frac{5 x ^{42}}{2} + \frac{35 x ^{39}}{3} + \frac{455 x ^{36}}{12} + 91 x^{33} + \frac{1001 x ^{30}}{6} + \frac{715 x ^{27}}{3} + \frac{2145 x ^{24}}{8} + \frac{715 x ^{21}}{3} + \frac{1001 x ^{18}}{6} + 91 x^{15} + \frac{455 x ^{12}}{12} + \frac{35 x ^{9}}{3} + \frac{5 x ^{6}}{2} + \frac{x ^{3}}{3} + C

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