integral of x^2(x^3+7)^{12}

解答

\int x^{2} (x^{3} + 7)^{12} d x

\frac{x ^{39}}{39} + \frac{7 x ^{36}}{3} + 98 x^{33} + \frac{7546 x ^{30}}{3} + \frac{132055 x ^{27}}{3} + 554631 x^{24} + 5176556 x^{21} + 36235892 x^{18} + 190238433 x^{15} + \frac{2219448385 x ^{12}}{3} + \frac{6214455478 x ^{9}}{3} + 7^{11} \cdot 2 x^{6} + \frac{7 ^{12} x ^{3}}{3} + C

求解步骤

\int x^{2} (x^{3} + 7)^{12} d x

乘开

x^{2} (x^{3} + 7)^{12} : x^{38} + 84 x^{35} + 3234 x^{32} + 75460 x^{29} + 1188495 x^{26} + 13311144 x^{23} + 108707676 x^{20} + 652246056 x^{17} + 2853576495 x^{14} + 8877793540 x^{11} + 18643366434 x^{8} + 7^{11} \cdot 12 x^{5} + 7^{12} x^{2}

= \int x^{38} + 84 x^{35} + 3234 x^{32} + 75460 x^{29} + 1188495 x^{26} + 13311144 x^{23} + 108707676 x^{20} + 652246056 x^{17} + 2853576495 x^{14} + 8877793540 x^{11} + 18643366434 x^{8} + 7^{11} \cdot 12 x^{5} + 7^{12} 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^{38} d x + \int 84 x^{35} d x + \int 3234 x^{32} d x + \int 75460 x^{29} d x + \int 1188495 x^{26} d x + \int 13311144 x^{23} d x + \int 108707676 x^{20} d x + \int 652246056 x^{17} d x + \int 2853576495 x^{14} d x + \int 8877793540 x^{11} d x + \int 18643366434 x^{8} d x + \int 7^{11} \cdot 12 x^{5} d x + \int 7^{12} x^{2} d x

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

\int 84 x^{35} d x = \frac{7 x ^{36}}{3}

\int 3234 x^{32} d x = 98 x^{33}

\int 75460 x^{29} d x = \frac{7546 x ^{30}}{3}

\int 1188495 x^{26} d x = \frac{132055 x ^{27}}{3}

\int 13311144 x^{23} d x = 554631 x^{24}

\int 108707676 x^{20} d x = 5176556 x^{21}

\int 652246056 x^{17} d x = 36235892 x^{18}

\int 2853576495 x^{14} d x = 190238433 x^{15}

\int 8877793540 x^{11} d x = \frac{2219448385 x ^{12}}{3}

\int 18643366434 x^{8} d x = \frac{6214455478 x ^{9}}{3}

\int 7^{11} \cdot 12 x^{5} d x = 7^{11} \cdot 2 x^{6}

\int 7^{12} x^{2} d x = \frac{7 ^{12} x ^{3}}{3}

= \frac{x ^{39}}{39} + \frac{7 x ^{36}}{3} + 98 x^{33} + \frac{7546 x ^{30}}{3} + \frac{132055 x ^{27}}{3} + 554631 x^{24} + 5176556 x^{21} + 36235892 x^{18} + 190238433 x^{15} + \frac{2219448385 x ^{12}}{3} + \frac{6214455478 x ^{9}}{3} + 7^{11} \cdot 2 x^{6} + \frac{7 ^{12} x ^{3}}{3}

解答补常数

= \frac{x ^{39}}{39} + \frac{7 x ^{36}}{3} + 98 x^{33} + \frac{7546 x ^{30}}{3} + \frac{132055 x ^{27}}{3} + 554631 x^{24} + 5176556 x^{21} + 36235892 x^{18} + 190238433 x^{15} + \frac{2219448385 x ^{12}}{3} + \frac{6214455478 x ^{9}}{3} + 7^{11} \cdot 2 x^{6} + \frac{7 ^{12} x ^{3}}{3} + C

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