What is the integral of xsin(1-x) ?

The integral of xsin(1-x) is xcos(1-x)+sin(1-x)+C

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integral of xsin(1-x)

\int x sin (1 - x) d x

x cos (1 - x) + sin (1 - x) + C

Solution steps

\int x sin (1 - x) d x

Apply u-substitution

: \int - (- u + 1) sin (u) d u

= \int - (- u + 1) sin (u) d u

Take the constant out:

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

= - \int (- u + 1) sin (u) d u

Apply Integration By Parts

: - cos (u) (- u + 1) - \int cos (u) d u

= - (- cos (u) (- u + 1) - \int cos (u) d u)

\int cos (u) d u = sin (u)

= - (- cos (u) (- u + 1) - sin (u))

Substitute back

u = 1 - x

= - (- cos (1 - x) (- (1 - x) + 1) - sin (1 - x))

Simplify

- (- cos (1 - x) (- (1 - x) + 1) - sin (1 - x)) : x cos (1 - x) + sin (1 - x)

= x cos (1 - x) + sin (1 - x)

Add a constant to the solution

= x cos (1 - x) + sin (1 - x) + C

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What is the integral of xsin(1-x) ?
The integral of xsin(1-x) is xcos(1-x)+sin(1-x)+C