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

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

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integral of e^{-x}x^3

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

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

Solution steps

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

y^{^{''}} + 4 y^{^{'}} + 9 y = 0, y (0) = 1, y^{^{'}} (0) = 13

9 (5^{x})

y^{^{'}} = 9 x y

x \to 2 - lim (x^{2} + 3)

\frac{d y}{d x} = (\frac{2 y + 1}{4 x - 3})^{2}

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