What is the integral from 4 to 6 of 0.2e^{-0.2x} ?

The integral from 4 to 6 of 0.2e^{-0.2x} is 0.14813…

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integral from 4 to 6 of 0.2e^{-0.2x}

\int_{4}^{6} 0.2 e^{- 0.2 x} d x

0.14813 \dots

Solution steps

\int_{4}^{6} 0.2 e^{- 0.2 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_{4}^{6} e^{- 0.2 x} d x

Apply u-substitution

= 0.2 \cdot \int_{- 0.8}^{- 1.2} - \frac{1}{0.2} e^{u} d u

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

= 0.2 (- \int_{- 1.2}^{- 0.8} - \frac{1}{0.2} 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}{0.2} \cdot \int_{- 1.2}^{- 0.8} e^{u} d u))

Simplify

= 0.2 (- (- 5 \cdot \int_{- 1.2}^{- 0.8} e^{u} d u))

Use the common integral:

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

= 0.2 (- (- 5 [e^{u}]_{- 1.2}^{- 0.8}))

Simplify

= [e^{u}]_{- 1.2}^{- 0.8}

Compute the boundaries:

0.14813 \dots

= 0.14813 \dots

x \to 3 lim (\frac{sin ( 3 x )}{x})

\int \frac{x}{( x ^{2} + 6 ) ^{2}} d x

\frac{\partial}{\partial x} (3 e^{s i n (x + 4 c t)})

\frac{( x + 1 ) d y}{d x} + (x + 2) y = 2 x e^{- x}

\int \frac{c}{x} d x

What is the integral from 4 to 6 of 0.2e^{-0.2x} ?
The integral from 4 to 6 of 0.2e^{-0.2x} is 0.14813…