What is the integral of (x+x^2)e^{-x} ?

The integral of (x+x^2)e^{-x} is -3e^{-x}x-3e^{-x}-e^{-x}x^2+C

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integral of (x+x^2)e^{-x}

\int (x + x^{2}) e^{- x} d x

- 3 e^{- x} x - 3 e^{- x} - e^{- x} x^{2} + C

Solution steps

\int (x + x^{2}) e^{- x} d x

Apply u-substitution

: \int - e^{u} (- u + u^{2}) d u

= \int - e^{u} (- u + u^{2}) d u

Take the constant out:

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

= - \int e^{u} (- u + u^{2}) d u

Expand

e^{u} (- u + u^{2}) : - e^{u} u + e^{u} u^{2}

= - \int - e^{u} u + e^{u} u^{2} 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 e^{u} u d u + \int e^{u} u^{2} d u)

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

\int e^{u} u^{2} d u = u^{2} e^{u} - 2 (e^{u} u - e^{u})

= - (- (e^{u} u - e^{u}) + u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - x

= - (- (e^{- x} (- x) - e^{- x}) + (- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x}))

Simplify

- (- (e^{- x} (- x) - e^{- x}) + (- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x})) : - 3 e^{- x} x - 3 e^{- x} - e^{- x} x^{2}

= - 3 e^{- x} x - 3 e^{- x} - e^{- x} x^{2}

Add a constant to the solution

= - 3 e^{- x} x - 3 e^{- x} - e^{- x} x^{2} + C

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What is the integral of (x+x^2)e^{-x} ?
The integral of (x+x^2)e^{-x} is -3e^{-x}x-3e^{-x}-e^{-x}x^2+C