What is the integral of t^2e^{-21t} ?

The integral of t^2e^{-21t} is -1/9261 (441e^{-21t}t^2-2(-21e^{-21t}t-e^{-21t}))+C

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integral of t^2e^{-21t}

\int t^{2} e^{- 21 t} d t

- \frac{1}{9261} (441 e^{- 21 t} t^{2} - 2 (- 21 e^{- 21 t} t - e^{- 21 t})) + C

Solution steps

\int t^{2} e^{- 21 t} d t

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{9261} d u

= \int - \frac{e ^{u} u ^{2}}{9261} d u

Take the constant out:

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

= - \frac{1}{9261} \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}{9261} (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}{9261} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 21 t

= - \frac{1}{9261} ((- 21 t)^{2} e^{- 21 t} - 2 (e^{- 21 t} (- 21 t) - e^{- 21 t}))

Simplify

- \frac{1}{9261} ((- 21 t)^{2} e^{- 21 t} - 2 (e^{- 21 t} (- 21 t) - e^{- 21 t})) : - \frac{1}{9261} (441 e^{- 21 t} t^{2} - 2 (- 21 e^{- 21 t} t - e^{- 21 t}))

= - \frac{1}{9261} (441 e^{- 21 t} t^{2} - 2 (- 21 e^{- 21 t} t - e^{- 21 t}))

Add a constant to the solution

= - \frac{1}{9261} (441 e^{- 21 t} t^{2} - 2 (- 21 e^{- 21 t} t - e^{- 21 t})) + C

y^{^{'}} = 3 x y^{2}

\frac{d}{d x} (\frac{x}{3 x ^{2} + 4})

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

2 x y^{^{'}} + y = 6 x, y (4) = 22

\int \frac{1}{( x + 2 ) ( x + 4 ) ( x + 6 )} d x

What is the integral of t^2e^{-21t} ?
The integral of t^2e^{-21t} is -1/9261 (441e^{-21t}t^2-2(-21e^{-21t}t-e^{-21t}))+C