What is the integral of-1/2 x^3e^{-x} ?

The integral of-1/2 x^3e^{-x} is -1/2 (-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-1/2 x^3e^{-x}

\int - \frac{1}{2} x^{3} e^{- x} d x

- \frac{1}{2} (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))) + C

Solution steps

\int - \frac{1}{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

= - \frac{1}{2} \cdot \int x^{3} e^{- x} d x

Apply u-substitution

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

= - \frac{1}{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

= - \frac{1}{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}))

= - \frac{1}{2} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = - x

= - \frac{1}{2} ((- x)^{3} e^{- x} - 3 ((- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x})))

Simplify

- \frac{1}{2} ((- x)^{3} e^{- x} - 3 ((- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x}))) : - \frac{1}{2} (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})))

= - \frac{1}{2} (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})))

Add a constant to the solution

= - \frac{1}{2} (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))) + C

\int \frac{v}{1 + v ^{2}} d v

\int 0.06 q - 0.6 q - \frac{7700}{q ^{2}} d q

\frac{d y}{d x} = a - b y^{2}

\int_{1}^{2} 4 x^{2} - 3 d x

x \to 2 - lim (k x^{2})

What is the integral of-1/2 x^3e^{-x} ?
The integral of-1/2 x^3e^{-x} is -1/2 (-e^{-x}x^3-3(e^{-x}x^2-2(-e^{-x}x-e^{-x})))+C