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

The integral of e^{-st}*e^{at} is 1/(-s+a)e^{-st+at}+C

Popular >

integral of e^{-st}*e^{at}

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

\frac{1}{- s + a} e^{- s t + a t} + C

Solution steps

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

Simplify

e^{- s t} e^{a t} : e^{- s t + a t}

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

Apply u-substitution

: \int \frac{e ^{u}}{- s + a} d u

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{- s + a} e^{u}

Substitute back

u = - s t + a t

= \frac{1}{- s + a} e^{- s t + a t}

Add a constant to the solution

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

f (x) = \frac{2}{3 x + 5}, (- 1, - 1)

f (x) = x^{2} - 2 x, at x = - 1

y^{^{'}} + 5 y = 2 sin (e^{5 x})

\int_{2}^{6} t^{3} ln (5 x) d x

\frac{1}{( s ^{2} - 1 )}

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