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

Lösung

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

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

Dezimale

0.21616 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= \frac{1}{2} \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

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Vereinfache

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

= \frac{1}{4} [e^{u}]_{- 2}^{0}

Berechne die Grenzen:

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

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

Vereinfache

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

x \to 0 lim (sinh (x) cot (x))

n = 0 \sum \infty n x^{n}

\frac{d y}{d x} + 12 y = 5

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

\int \frac{x ^{3}}{49 + x ^{2}} d x