What is the integral of 5sin(x-6)+54x^8+9sin(x-6) ?

The integral of 5sin(x-6)+54x^8+9sin(x-6) is 14(xsin(x-6)+xsin(6-x)-cos(6-x))+6x^9+C

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

\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

Solution steps

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

Apply the Sum Rule:

\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))

Simplify

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

Add a constant to the solution

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

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What is the integral of 5sin(x-6)+54x^8+9sin(x-6) ?
The integral of 5sin(x-6)+54x^8+9sin(x-6) is 14(xsin(x-6)+xsin(6-x)-cos(6-x))+6x^9+C