integral of x^2(7x^3-9)^5

解答

\int x^{2} (7 x^{3} - 9)^{5} d x

\frac{16807 x ^{18}}{18} - 7203 x^{15} + \frac{46305 x ^{12}}{2} - 39690 x^{9} + \frac{76545 x ^{6}}{2} - 19683 x^{3} + C

求解步骤

\int x^{2} (7 x^{3} - 9)^{5} d x

乘开

x^{2} (7 x^{3} - 9)^{5} : 16807 x^{17} - 108045 x^{14} + 277830 x^{11} - 357210 x^{8} + 229635 x^{5} - 59049 x^{2}

= \int 16807 x^{17} - 108045 x^{14} + 277830 x^{11} - 357210 x^{8} + 229635 x^{5} - 59049 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 16807 x^{17} d x - \int 108045 x^{14} d x + \int 277830 x^{11} d x - \int 357210 x^{8} d x + \int 229635 x^{5} d x - \int 59049 x^{2} d x

\int 16807 x^{17} d x = \frac{16807 x ^{18}}{18}

\int 108045 x^{14} d x = 7203 x^{15}

\int 277830 x^{11} d x = \frac{46305 x ^{12}}{2}

\int 357210 x^{8} d x = 39690 x^{9}

\int 229635 x^{5} d x = \frac{76545 x ^{6}}{2}

\int 59049 x^{2} d x = 19683 x^{3}

= \frac{16807 x ^{18}}{18} - 7203 x^{15} + \frac{46305 x ^{12}}{2} - 39690 x^{9} + \frac{76545 x ^{6}}{2} - 19683 x^{3}

解答补常数

= \frac{16807 x ^{18}}{18} - 7203 x^{15} + \frac{46305 x ^{12}}{2} - 39690 x^{9} + \frac{76545 x ^{6}}{2} - 19683 x^{3} + C

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