What is the integral of-te^{-st} ?

The integral of-te^{-st} is -1/(s^2)(-se^{-st}t-e^{-st})+C

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integral of-te^{-st}

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

- \frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

Solution steps

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

Take the constant out:

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

= - \int e^{- s t} t d t

Apply u-substitution

: \int \frac{e ^{u} u}{s ^{2}} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = - s t

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

Simplify

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

Add a constant to the solution

= - \frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

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\int y^{\frac{1}{7}} d y

\frac{d}{d θ} (\frac{2}{sin ( θ )})

What is the integral of-te^{-st} ?
The integral of-te^{-st} is -1/(s^2)(-se^{-st}t-e^{-st})+C