What is the integral of 2x^3*e^{-x} ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2x^3*e^{-x}

integral

2 x^{3} \cdot e^{- x}

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

Solution steps

\int 2 x^{3} e^{- x} d x

Take the constant out:

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

= 2 \cdot \int x^{3} e^{- x} d x

Apply u-substitution

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

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

Apply Integration By Parts

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

= 2 (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}))

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

Substitute back

u = - x

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

Simplify

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

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

Add a constant to the solution

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

\frac{y}{x ^{2} + y ^{2}}

2 + x

\frac{1}{( x ^{\frac{1}{2}} + 1 )}

(x^{2} - x)^{4}

\frac{2}{( 2 x + 3 )}

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