What is the integral of 5e^{-5t} ?

The integral of 5e^{-5t} is -e^{-5t}+C

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integral of 5e^{-5t}

\int 5 e^{- 5 t} d t

- e^{- 5 t} + C

Solution steps

\int 5 e^{- 5 t} d t

Take the constant out:

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

= 5 \cdot \int e^{- 5 t} d t

Apply u-substitution

: \int - \frac{1}{5} e^{u} d u

= 5 \cdot \int - \frac{1}{5} e^{u} d u

Take the constant out:

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

= 5 (- \frac{1}{5} \cdot \int e^{u} d u)

Use the common integral:

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

= 5 (- \frac{1}{5} e^{u})

Substitute back

u = - 5 t

= 5 (- \frac{1}{5} e^{- 5 t})

Simplify

5 (- \frac{1}{5} e^{- 5 t}) : - e^{- 5 t}

= - e^{- 5 t}

Add a constant to the solution

= - e^{- 5 t} + C

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