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

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

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integral of x^3*e^{-x}

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

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

Solution steps

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

Apply u-substitution

: \int e^{u} u^{3} d u

= \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= u^{3} e^{u} - \int 3 u^{2} e^{u} d u

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - x

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

Simplify

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

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

Add a constant to the solution

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

f (x) = \frac{5}{x}, (- 3, (- \frac{5}{3}))

\frac{d x}{d t} + 3 = 0

y = 4 + csc (t^{4})

y = (\frac{x - 1}{x ^{2} + x + 1})^{4}

y^{^{'}} + 2 x y = 2 x

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