What is the integral of xe^{3-x} ?

The integral of xe^{3-x} is -e^{3-x}x-e^{3-x}+C

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integral of xe^{3-x}

\int x e^{3 - x} d x

- e^{3 - x} x - e^{3 - x} + C

Solution steps

\int x e^{3 - x} d x

Apply u-substitution

: \int - e^{u} (- u + 3) d u

= \int - e^{u} (- u + 3) d u

Take the constant out:

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

= - \int e^{u} (- u + 3) d u

Apply Integration By Parts

: e^{u} (- u + 3) - \int - e^{u} d u

= - (e^{u} (- u + 3) - \int - e^{u} d u)

\int - e^{u} d u = - e^{u}

= - (e^{u} (- u + 3) - (- e^{u}))

Substitute back

u = 3 - x

= - (e^{3 - x} (- (3 - x) + 3) - (- e^{3 - x}))

Simplify

- (e^{3 - x} (- (3 - x) + 3) - (- e^{3 - x})) : - e^{3 - x} x - e^{3 - x}

= - e^{3 - x} x - e^{3 - x}

Add a constant to the solution

= - e^{3 - x} x - e^{3 - x} + C

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What is the integral of xe^{3-x} ?
The integral of xe^{3-x} is -e^{3-x}x-e^{3-x}+C