integral of cos(7x)(11+sin(7x))^5

解答

\int cos (7 x) (11 + sin (7 x))^{5} d x

\frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x) + C

求解步骤

\int cos (7 x) (11 + sin (7 x))^{5} d x

乘开

cos (7 x) (11 + sin (7 x))^{5} : 161051 cos (7 x) + 73205 cos (7 x) sin (7 x) + 13310 sin^{2} (7 x) cos (7 x) + 1210 sin^{3} (7 x) cos (7 x) + 55 sin^{4} (7 x) cos (7 x) + sin^{5} (7 x) cos (7 x)

= \int 161051 cos (7 x) + 73205 cos (7 x) sin (7 x) + 13310 sin^{2} (7 x) cos (7 x) + 1210 sin^{3} (7 x) cos (7 x) + 55 sin^{4} (7 x) cos (7 x) + sin^{5} (7 x) cos (7 x) 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 161051 cos (7 x) d x + \int 73205 cos (7 x) sin (7 x) d x + \int 13310 sin^{2} (7 x) cos (7 x) d x + \int 1210 sin^{3} (7 x) cos (7 x) d x + \int 55 sin^{4} (7 x) cos (7 x) d x + \int sin^{5} (7 x) cos (7 x) d x

\int 161051 cos (7 x) d x = \frac{161051}{7} sin (7 x)

\int 73205 cos (7 x) sin (7 x) d x = - \frac{73205}{14} cos^{2} (7 x)

\int 13310 sin^{2} (7 x) cos (7 x) d x = \frac{13310}{21} sin^{3} (7 x)

\int 1210 sin^{3} (7 x) cos (7 x) d x = \frac{605}{14} sin^{4} (7 x)

\int 55 sin^{4} (7 x) cos (7 x) d x = \frac{11}{7} sin^{5} (7 x)

\int sin^{5} (7 x) cos (7 x) d x = \frac{1}{42} sin^{6} (7 x)

= \frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x)

解答补常数

= \frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x) + C

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