What is the integral of te^{-7t} ?

The integral of te^{-7t} is 1/49 (-7e^{-7t}t-e^{-7t})+C

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integral of te^{-7t}

\int t e^{- 7 t} d t

\frac{1}{49} (- 7 e^{- 7 t} t - e^{- 7 t}) + C

Solution steps

\int t e^{- 7 t} d t

Apply u-substitution

= \int \frac{e ^{u} u}{49} d u

Take the constant out:

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

= \frac{1}{49} \cdot \int e^{u} u d u

Apply Integration By Parts

= \frac{1}{49} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{49} (e^{u} u - e^{u})

Substitute back

u = - 7 t

= \frac{1}{49} (e^{- 7 t} (- 7 t) - e^{- 7 t})

Simplify

= \frac{1}{49} (- 7 e^{- 7 t} t - e^{- 7 t})

Add a constant to the solution

= \frac{1}{49} (- 7 e^{- 7 t} t - e^{- 7 t}) + C

What is the integral of te^{-7t} ?
The integral of te^{-7t} is 1/49 (-7e^{-7t}t-e^{-7t})+C