What is the integral of 0.4*e^{-2000x} ?

The integral of 0.4*e^{-2000x} is -0.0002e^{-2000x}+C

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integral of 0.4*e^{-2000x}

\int 0.4 \cdot e^{- 2000 x} d x

- 0.0002 e^{- 2000 x} + C

Solution steps

\int 0.4 e^{- 2000 x} d x

Take the constant out:

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

= 0.4 \cdot \int e^{- 2000 x} d x

Apply u-substitution

: \int - \frac{1}{2000} e^{u} d u

= 0.4 \cdot \int - \frac{1}{2000} e^{u} d u

Take the constant out:

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

= 0.4 (- \frac{1}{2000} \cdot \int e^{u} d u)

Use the common integral:

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

= 0.4 (- \frac{1}{2000} e^{u})

Substitute back

u = - 2000 x

= 0.4 (- \frac{1}{2000} e^{- 2000 x})

Simplify

0.4 (- \frac{1}{2000} e^{- 2000 x}) : - 0.0002 e^{- 2000 x}

= - 0.0002 e^{- 2000 x}

Add a constant to the solution

= - 0.0002 e^{- 2000 x} + C

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What is the integral of 0.4*e^{-2000x} ?
The integral of 0.4*e^{-2000x} is -0.0002e^{-2000x}+C