What is the integral of e^{-st}(3t) ?

The integral of e^{-st}(3t) is 3/(s^2)(-se^{-st}t-e^{-st})+C

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integral of e^{-st}(3t)

\int e^{- s t} (3 t) d t

\frac{3}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

Solution steps

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

Apply u-substitution

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

= 3 \cdot \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

= 3 \cdot \frac{1}{s ^{2}} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 3 \cdot \frac{1}{s ^{2}} (e^{u} u - \int e^{u} d u)

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

= 3 \cdot \frac{1}{s ^{2}} (e^{u} u - e^{u})

Substitute back

u = - s t

= 3 \cdot \frac{1}{s ^{2}} (e^{- s t} (- s t) - e^{- s t})

Simplify

3 \cdot \frac{1}{s ^{2}} (e^{- s t} (- s t) - e^{- s t}) : \frac{3}{s ^{2}} (- s e^{- s t} t - e^{- s t})

= \frac{3}{s ^{2}} (- s e^{- s t} t - e^{- s t})

Add a constant to the solution

= \frac{3}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

\int 5 x (x - x^{2}) d x

\frac{\partial}{\partial x} (ln (x^{2} + y) - z)

x \to 3 lim (x^{2} + 9)

2 sin^{2} (x)

f (y) = \frac{1}{2 y}

What is the integral of e^{-st}(3t) ?
The integral of e^{-st}(3t) is 3/(s^2)(-se^{-st}t-e^{-st})+C