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

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

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integral of Ce^{-at}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= C \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

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

Use the common integral:

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

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

Substitute back

u = - a t

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

Simplify

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

Add a constant to the solution

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

f (x) = \frac{6}{x ^{4}}

y^{^{'}} = \frac{x + 2 y}{x}

\int e^{- l n (x)} d x

x = y^{2}, x = y + 2