What is the integral of 0.2e^{-0.3x} ?

The integral of 0.2e^{-0.3x} is -0.66666…e^{-0.3x}+C

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integral of 0.2e^{-0.3x}

\int 0.2 e^{- 0.3 x} d x

- 0.66666 \dots e^{- 0.3 x} + C

Solution steps

\int 0.2 e^{- 0.3 x} d x

Take the constant out:

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

= 0.2 \cdot \int e^{- 0.3 x} d x

Apply u-substitution

: \int - \frac{1}{0.3} e^{u} d u

= 0.2 \cdot \int - \frac{1}{0.3} e^{u} d u

Take the constant out:

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

= 0.2 (- \frac{1}{0.3} \cdot \int e^{u} d u)

Simplify

= 0.2 (- 3.33333 \dots \cdot \int e^{u} d u)

Use the common integral:

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

= 0.2 (- 3.33333 \dots e^{u})

Substitute back

u = - 0.3 x

= 0.2 (- 3.33333 \dots e^{- 0.3 x})

Simplify

= - 0.66666 \dots e^{- 0.3 x}

Add a constant to the solution

= - 0.66666 \dots e^{- 0.3 x} + C

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What is the integral of 0.2e^{-0.3x} ?
The integral of 0.2e^{-0.3x} is -0.66666…e^{-0.3x}+C