integral of 5sin(x-6)+54x^8+9sin(x-6)

解答

\int 5 sin (x - 6) + 54 x^{8} + 9 sin (x - 6) d x

14 (x sin (x - 6) + x sin (6 - x) - cos (6 - x)) + 6 x^{9} + C

求解步骤

\int 5 sin (x - 6) + 54 x^{8} + 9 sin (x - 6) 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 5 sin (x - 6) d x + \int 54 x^{8} d x + \int 9 sin (x - 6) d x

\int 5 sin (x - 6) d x = 5 (x sin (x - 6) + x sin (6 - x) - cos (6 - x))

\int 54 x^{8} d x = 6 x^{9}

\int 9 sin (x - 6) d x = 9 (x sin (x - 6) + x sin (6 - x) - cos (6 - x))

= 5 (x sin (x - 6) + x sin (6 - x) - cos (6 - x)) + 6 x^{9} + 9 (x sin (x - 6) + x sin (6 - x) - cos (6 - x))

化简

= 14 (x sin (x - 6) + x sin (6 - x) - cos (6 - x)) + 6 x^{9}

解答补常数

= 14 (x sin (x - 6) + x sin (6 - x) - cos (6 - x)) + 6 x^{9} + C

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