What is the integral from 0 to 1 of 8e^{-2x} ?

The integral from 0 to 1 of 8e^{-2x} is 4(1-1/(e^2))

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integral from 0 to 1 of 8e^{-2x}

\int_{0}^{1} 8 e^{- 2 x} d x

4 (1 - \frac{1}{e ^{2}})

Solution steps

\int_{0}^{1} 8 e^{- 2 x} d x

Take the constant out:

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

= 8 \cdot \int_{0}^{1} e^{- 2 x} d x

Apply u-substitution

: \int_{0}^{- 2} - \frac{1}{2} e^{u} d u

= 8 \cdot \int_{0}^{- 2} - \frac{1}{2} e^{u} d u

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

= 8 (- \int_{- 2}^{0} - \frac{1}{2} e^{u} d u)

Take the constant out:

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

= 8 (- (- \frac{1}{2} \cdot \int_{- 2}^{0} e^{u} d u))

Use the common integral:

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

= 8 (- (- \frac{1}{2} [e^{u}]_{- 2}^{0}))

Simplify

8 (- (- \frac{1}{2} [e^{u}]_{- 2}^{0})) : 4 [e^{u}]_{- 2}^{0}

= 4 [e^{u}]_{- 2}^{0}

Compute the boundaries:

1 - \frac{1}{e ^{2}}

= 4 (1 - \frac{1}{e ^{2}})

x \to 8 lim (x - 8)

\frac{10 s - 3}{s ^{2} + 4 s + 29}

f (x) = sin (x), at x = \frac{5 π}{2}

\frac{1}{3 - x}

\int_{1}^{e} \frac{x ^{2} + x + 1}{x} d x

What is the integral from 0 to 1 of 8e^{-2x} ?
The integral from 0 to 1 of 8e^{-2x} is 4(1-1/(e^2))