What is the integral from x to 0 of e^{-3t} ?

The integral from x to 0 of e^{-3t} is -1/3 (1-e^{-3x})

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integral from x to 0 of e^{-3t}

\int_{x}^{0} e^{- 3 t} d t

- \frac{1}{3} (1 - e^{- 3 x})

Solution steps

\int_{x}^{0} e^{- 3 t} d t

Apply u-substitution

= \int_{- 3 x}^{0} - \frac{1}{3} e^{u} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int_{- 3 x}^{0} e^{u} d u

Use the common integral:

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

= - \frac{1}{3} [e^{u}]_{- 3 x}^{0}

Compute the boundaries:

1 - e^{- 3 x}

= - \frac{1}{3} (1 - e^{- 3 x})

What is the integral from x to 0 of e^{-3t} ?
The integral from x to 0 of e^{-3t} is -1/3 (1-e^{-3x})