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

The integral of e^{-at} is -1/a e^{-at}+C

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integral of e^{-at}

\int e^{- a t} d t

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

Solution steps

\int e^{- a t} d t

Apply u-substitution

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

= \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

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

Use the common integral:

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

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

Substitute back

u = - a t

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{x + y}{x + y + 1}

x \to \infty lim (1.3 x)

\int 7 sin (8 x) cos (5 x) d x

\frac{2}{s} - \frac{1}{s ^{3}}

sin (π x), 0.5