What is the integral of 0.1e^{-0.1x} ?

The integral of 0.1e^{-0.1x} is -e^{-0.1x}+C

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integral of 0.1e^{-0.1x}

\int 0.1 e^{- 0.1 x} d x

- e^{- 0.1 x} + C

Solution steps

\int 0.1 e^{- 0.1 x} d x

Take the constant out:

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

= 0.1 \cdot \int e^{- 0.1 x} d x

Apply u-substitution

: \int - \frac{1}{0.1} e^{u} d u

= 0.1 \cdot \int - \frac{1}{0.1} e^{u} d u

Take the constant out:

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

= 0.1 (- \frac{1}{0.1} \cdot \int e^{u} d u)

Simplify

= 0.1 (- 10 \cdot \int e^{u} d u)

Use the common integral:

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

= 0.1 (- 10 e^{u})

Substitute back

u = - 0.1 x

= 0.1 (- 10 e^{- 0.1 x})

Simplify

= - e^{- 0.1 x}

Add a constant to the solution

= - e^{- 0.1 x} + C

\frac{d y}{d t} = sin (3 t) - 2 y

x \to 9 + lim (\frac{5}{x - 9})

\int (5 + 3 x + 1) d x

3 sec (x)

y^{^{'}} + y = sin (t)

What is the integral of 0.1e^{-0.1x} ?
The integral of 0.1e^{-0.1x} is -e^{-0.1x}+C