What is the integral of ae^{-at} ?

The integral of ae^{-at} is -e^{-at}+C

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integral of ae^{-at}

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

- e^{- a t} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = - a t

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

Simplify

a (- \frac{1}{a} e^{- a t}) : - e^{- a t}

= - e^{- a t}

Add a constant to the solution

= - e^{- a t} + C