What is the integral of t^2e^{-11t} ?

The integral of t^2e^{-11t} is -1/1331 (121e^{-11t}t^2-2(-11e^{-11t}t-e^{-11t}))+C

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integral of t^2e^{-11t}

\int t^{2} e^{- 11 t} d t

- \frac{1}{1331} (121 e^{- 11 t} t^{2} - 2 (- 11 e^{- 11 t} t - e^{- 11 t})) + C

Solution steps

\int t^{2} e^{- 11 t} d t

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{1331} d u

= \int - \frac{e ^{u} u ^{2}}{1331} d u

Take the constant out:

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

= - \frac{1}{1331} \cdot \int e^{u} u^{2} d u

Apply Integration By Parts

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

= - \frac{1}{1331} (u^{2} e^{u} - \int 2 u e^{u} d u)

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

= - \frac{1}{1331} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 11 t

= - \frac{1}{1331} ((- 11 t)^{2} e^{- 11 t} - 2 (e^{- 11 t} (- 11 t) - e^{- 11 t}))

Simplify

- \frac{1}{1331} ((- 11 t)^{2} e^{- 11 t} - 2 (e^{- 11 t} (- 11 t) - e^{- 11 t})) : - \frac{1}{1331} (121 e^{- 11 t} t^{2} - 2 (- 11 e^{- 11 t} t - e^{- 11 t}))

= - \frac{1}{1331} (121 e^{- 11 t} t^{2} - 2 (- 11 e^{- 11 t} t - e^{- 11 t}))

Add a constant to the solution

= - \frac{1}{1331} (121 e^{- 11 t} t^{2} - 2 (- 11 e^{- 11 t} t - e^{- 11 t})) + C

x \to 2 lim (3^{x + 1})

x \to 9 lim (\frac{4 x - 3}{11})

x \to c lim (f (c))

\int 1 arcsin (x) d x

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

What is the integral of t^2e^{-11t} ?
The integral of t^2e^{-11t} is -1/1331 (121e^{-11t}t^2-2(-11e^{-11t}t-e^{-11t}))+C