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受欢迎的微积分 >

integral of-sin(3x)(5+cos(3x))^4

解答

\int - sin (3 x) (5 + cos (3 x))^{4} d x

解答

\frac{625}{3} cos (3 x) + \frac{250}{3} cos^{2} (3 x) + \frac{50}{3} cos^{3} (3 x) + \frac{5}{3} cos^{4} (3 x) + \frac{1}{15} cos^{5} (3 x) + C

求解步骤

\int - sin (3 x) (5 + cos (3 x))^{4} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int sin (3 x) (5 + cos (3 x))^{4} d x

乘开

sin (3 x) (5 + cos (3 x))^{4} : 625 sin (3 x) + 500 sin (3 x) cos (3 x) + 150 cos^{2} (3 x) sin (3 x) + 20 cos^{3} (3 x) sin (3 x) + cos^{4} (3 x) sin (3 x)

= - \int 625 sin (3 x) + 500 sin (3 x) cos (3 x) + 150 cos^{2} (3 x) sin (3 x) + 20 cos^{3} (3 x) sin (3 x) + cos^{4} (3 x) sin (3 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 625 sin (3 x) d x + \int 500 sin (3 x) cos (3 x) d x + \int 150 cos^{2} (3 x) sin (3 x) d x + \int 20 cos^{3} (3 x) sin (3 x) d x + \int cos^{4} (3 x) sin (3 x) d x)

\int 625 sin (3 x) d x = - \frac{625}{3} cos (3 x)

\int 500 sin (3 x) cos (3 x) d x = - \frac{250}{3} cos^{2} (3 x)

\int 150 cos^{2} (3 x) sin (3 x) d x = - \frac{50}{3} cos^{3} (3 x)

\int 20 cos^{3} (3 x) sin (3 x) d x = - \frac{5}{3} cos^{4} (3 x)

\int cos^{4} (3 x) sin (3 x) d x = - \frac{1}{15} cos^{5} (3 x)

= - (- \frac{625}{3} cos (3 x) - \frac{250}{3} cos^{2} (3 x) - \frac{50}{3} cos^{3} (3 x) - \frac{5}{3} cos^{4} (3 x) - \frac{1}{15} cos^{5} (3 x))

化简

= \frac{625}{3} cos (3 x) + \frac{250}{3} cos^{2} (3 x) + \frac{50}{3} cos^{3} (3 x) + \frac{5}{3} cos^{4} (3 x) + \frac{1}{15} cos^{5} (3 x)

解答补常数

= \frac{625}{3} cos (3 x) + \frac{250}{3} cos^{2} (3 x) + \frac{50}{3} cos^{3} (3 x) + \frac{5}{3} cos^{4} (3 x) + \frac{1}{15} cos^{5} (3 x) + C

作图

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