What is the integral of e^{-st}*t ?

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

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integral of e^{-st}*t

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

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

Solution steps

\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

9 arcsec (x)

\frac{\partial}{\partial y} (x y (\frac{1}{y ^{3}} + 2))

\int \frac{1}{4} e^{4 x + 1} d x

- x^{2} + 80 x - 600

sin^{2} (π x)

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