What is the integral of 5e^{-0.7t} ?

The integral of 5e^{-0.7t} is -7.14285…e^{-0.7t}+C

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integral of 5e^{-0.7t}

\int 5 e^{- 0.7 t} d t

- 7.14285 \dots e^{- 0.7 t} + C

Solution steps

\int 5 e^{- 0.7 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^{- 0.7 t} d t

Apply u-substitution

: \int - \frac{1}{0.7} e^{u} d u

= 5 \cdot \int - \frac{1}{0.7} 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}{0.7} \cdot \int e^{u} d u)

Simplify

= 5 (- 1.42857 \dots \cdot \int e^{u} d u)

Use the common integral:

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

= 5 (- 1.42857 \dots e^{u})

Substitute back

u = - 0.7 t

= 5 (- 1.42857 \dots e^{- 0.7 t})

Simplify

= - 7.14285 \dots e^{- 0.7 t}

Add a constant to the solution

= - 7.14285 \dots e^{- 0.7 t} + C

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What is the integral of 5e^{-0.7t} ?
The integral of 5e^{-0.7t} is -7.14285…e^{-0.7t}+C