integral of x^2(7x^3-3)^8

解答

\int x^{2} (7 x^{3} - 3)^{8} d x

\frac{5764801 x ^{27}}{27} - 823543 x^{24} + 1411788 x^{21} - 1411788 x^{18} + 907578 x^{15} - 388962 x^{12} + 111132 x^{9} - 20412 x^{6} + 2187 x^{3} + C

求解步骤

\int x^{2} (7 x^{3} - 3)^{8} d x

因式分解

7 x^{3} - 3 : - (- 7 x^{3} + 3)

= \int x^{2} (- (- 7 x^{3} + 3))^{8} d x

(- (- 7 x^{3} + 3))^{8} = (- 7 x^{3} + 3)^{8}

= \int x^{2} (- 7 x^{3} + 3)^{8} d x

乘开

x^{2} (- 7 x^{3} + 3)^{8} : 5764801 x^{26} - 19765032 x^{23} + 29647548 x^{20} - 25412184 x^{17} + 13613670 x^{14} - 4667544 x^{11} + 1000188 x^{8} - 122472 x^{5} + 6561 x^{2}

= \int 5764801 x^{26} - 19765032 x^{23} + 29647548 x^{20} - 25412184 x^{17} + 13613670 x^{14} - 4667544 x^{11} + 1000188 x^{8} - 122472 x^{5} + 6561 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 5764801 x^{26} d x - \int 19765032 x^{23} d x + \int 29647548 x^{20} d x - \int 25412184 x^{17} d x + \int 13613670 x^{14} d x - \int 4667544 x^{11} d x + \int 1000188 x^{8} d x - \int 122472 x^{5} d x + \int 6561 x^{2} d x

\int 5764801 x^{26} d x = \frac{5764801 x ^{27}}{27}

\int 19765032 x^{23} d x = 823543 x^{24}

\int 29647548 x^{20} d x = 1411788 x^{21}

\int 25412184 x^{17} d x = 1411788 x^{18}

\int 13613670 x^{14} d x = 907578 x^{15}

\int 4667544 x^{11} d x = 388962 x^{12}

\int 1000188 x^{8} d x = 111132 x^{9}

\int 122472 x^{5} d x = 20412 x^{6}

\int 6561 x^{2} d x = 2187 x^{3}

= \frac{5764801 x ^{27}}{27} - 823543 x^{24} + 1411788 x^{21} - 1411788 x^{18} + 907578 x^{15} - 388962 x^{12} + 111132 x^{9} - 20412 x^{6} + 2187 x^{3}

解答补常数

= \frac{5764801 x ^{27}}{27} - 823543 x^{24} + 1411788 x^{21} - 1411788 x^{18} + 907578 x^{15} - 388962 x^{12} + 111132 x^{9} - 20412 x^{6} + 2187 x^{3} + C

\int \frac{1}{1 + cos ( x ) + 7 sin ( x )} d x

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