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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of t^3e^{-t}

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

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

Solution steps

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

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 = - t

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

Simplify

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

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

Add a constant to the solution

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

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

x \to - 1 lim (\frac{2 - 2}{x + 1})

sin (\frac{10 π}{x})

\frac{d}{d t} (- cot (t))

x \to 2 lim (\frac{2 x + 1}{5 - 3 x})

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