What is the integral of (x^3-1)sin(pi/2-x) ?

The integral of (x^3-1)sin(pi/2-x) is

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integral of (x^3-1)sin(pi/2-x)

\int (x^{3} - 1) sin (\frac{π}{2} - x) d x

sin (\frac{π}{2} - x) - arcsin^{3} (sin (\frac{π}{2} - x)) sin (\frac{π}{2} - x) - 3 arcsin^{2} (sin (\frac{π}{2} - x)) cos (\frac{π}{2} - x) + 6 sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 6 cos (\frac{π}{2} - x) + C

Solution steps

\int (x^{3} - 1) sin (\frac{π}{2} - x) d x

Apply u-substitution

: \int 1 - arcsin^{3} (u) d u

= \int 1 - arcsin^{3} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int arcsin^{3} (u) d u

\int 1 d u = u

\int arcsin^{3} (u) d u = u arcsin^{3} (u) - 3 (- arcsin^{2} (u) 1 - u^{2} + 2 (u arcsin (u) + 1 - u^{2}))

= u - (u arcsin^{3} (u) - 3 (- arcsin^{2} (u) 1 - u^{2} + 2 (u arcsin (u) + 1 - u^{2})))

Substitute back

u = sin (\frac{π}{2} - x)

= sin (\frac{π}{2} - x) - (sin (\frac{π}{2} - x) arcsin^{3} (sin (\frac{π}{2} - x)) - 3 (- arcsin^{2} (sin (\frac{π}{2} - x)) 1 - sin^{2} (\frac{π}{2} - x) + 2 (sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 1 - sin^{2} (\frac{π}{2} - x))))

Simplify

sin (\frac{π}{2} - x) - (sin (\frac{π}{2} - x) arcsin^{3} (sin (\frac{π}{2} - x)) - 3 (- arcsin^{2} (sin (\frac{π}{2} - x)) 1 - sin^{2} (\frac{π}{2} - x) + 2 (sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 1 - sin^{2} (\frac{π}{2} - x)))) : sin (\frac{π}{2} - x) - arcsin^{3} (sin (\frac{π}{2} - x)) sin (\frac{π}{2} - x) - 3 arcsin^{2} (sin (\frac{π}{2} - x)) cos (\frac{π}{2} - x) + 6 sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 6 cos (\frac{π}{2} - x)

= sin (\frac{π}{2} - x) - arcsin^{3} (sin (\frac{π}{2} - x)) sin (\frac{π}{2} - x) - 3 arcsin^{2} (sin (\frac{π}{2} - x)) cos (\frac{π}{2} - x) + 6 sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 6 cos (\frac{π}{2} - x)

Add a constant to the solution

= sin (\frac{π}{2} - x) - arcsin^{3} (sin (\frac{π}{2} - x)) sin (\frac{π}{2} - x) - 3 arcsin^{2} (sin (\frac{π}{2} - x)) cos (\frac{π}{2} - x) + 6 sin (\frac{π}{2} - x) arcsin (sin (\frac{π}{2} - x)) + 6 cos (\frac{π}{2} - x) + C

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What is the integral of (x^3-1)sin(pi/2-x) ?
The integral of (x^3-1)sin(pi/2-x) is