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

The integral of 3te^{-3t} is -e^{-3t}t-1/3 e^{-3t}+C

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integral of 3te^{-3t}

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

- e^{- 3 t} t - \frac{1}{3} e^{- 3 t} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 3 \cdot \int \frac{e ^{u} u}{9} d u

Take the constant out:

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

= 3 \cdot \frac{1}{9} \cdot \int e^{u} u d u

Apply Integration By Parts

= 3 \cdot \frac{1}{9} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= 3 \cdot \frac{1}{9} (e^{u} u - e^{u})

Substitute back

u = - 3 t

= 3 \cdot \frac{1}{9} (e^{- 3 t} (- 3 t) - e^{- 3 t})

Simplify

3 \cdot \frac{1}{9} (e^{- 3 t} (- 3 t) - e^{- 3 t}) : - e^{- 3 t} t - \frac{1}{3} e^{- 3 t}

= - e^{- 3 t} t - \frac{1}{3} e^{- 3 t}

Add a constant to the solution

= - e^{- 3 t} t - \frac{1}{3} e^{- 3 t} + C

What is the integral of 3te^{-3t} ?
The integral of 3te^{-3t} is -e^{-3t}t-1/3 e^{-3t}+C