What is the integral from 0 to 3 of 3e^{-0.4t} ?

The integral from 0 to 3 of 3e^{-0.4t} is 5.24104…

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integral from 0 to 3 of 3e^{-0.4t}

\int_{0}^{3} 3 e^{- 0.4 t} d t

5.24104 \dots

Solution steps

\int_{0}^{3} 3 e^{- 0.4 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}^{3} e^{- 0.4 t} d t

Apply u-substitution

: \int_{0}^{- 1.2} - \frac{1}{0.4} e^{u} d u

= 3 \cdot \int_{0}^{- 1.2} - \frac{1}{0.4} e^{u} d u

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

= 3 (- \int_{- 1.2}^{0} - \frac{1}{0.4} 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}{0.4} \cdot \int_{- 1.2}^{0} e^{u} d u))

Simplify

= 3 (- (- 2.5 \cdot \int_{- 1.2}^{0} e^{u} d u))

Use the common integral:

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

= 3 (- (- 2.5 [e^{u}]_{- 1.2}^{0}))

Simplify

= 7.5 [e^{u}]_{- 1.2}^{0}

Compute the boundaries:

0.69880 \dots

= 7.5 \cdot 0.69880 \dots

Simplify

= 5.24104 \dots

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What is the integral from 0 to 3 of 3e^{-0.4t} ?
The integral from 0 to 3 of 3e^{-0.4t} is 5.24104…