What is the integral from 0 to 1 of 3e^{-3t} ?

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

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integral from 0 to 1 of 3e^{-3t}

\int_{0}^{1} 3 e^{- 3 t} d t

1 - \frac{1}{e ^{3}}

Decimal

0.95021 \dots

Solution steps

\int_{0}^{1} 3 e^{- 3 t} d t

Take the constant out:

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

= 3 \cdot \int_{0}^{1} e^{- 3 t} d t

Apply u-substitution

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 3 (- \int_{- 3}^{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

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

Use the common integral:

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

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

Simplify

3 (- (- \frac{1}{3} [e^{u}]_{- 3}^{0})) : [e^{u}]_{- 3}^{0}

= [e^{u}]_{- 3}^{0}

Compute the boundaries:

1 - \frac{1}{e ^{3}}

= 1 - \frac{1}{e ^{3}}

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