What is the integral of λe^{-λt} ?

The integral of λe^{-λt} is -e^{-λt}+C

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integral of λe^{-λt}

\int λ e^{- λ t} d t

- e^{- λ t} + C

Solution steps

\int λ e^{- λ t} d t

Take the constant out:

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

= λ \cdot \int e^{- λ t} d t

Apply u-substitution

= λ \cdot \int - \frac{e ^{u}}{λ} d u

Take the constant out:

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

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

Use the common integral:

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

= λ (- \frac{1}{λ} e^{u})

Substitute back

u = - λ t

= λ (- \frac{1}{λ} e^{- λ t})

Simplify

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

= - e^{- λ t}

Add a constant to the solution

= - e^{- λ t} + C