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

Lösung

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

1 - \frac{1}{e}

Dezimale

0.39346 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Vereinfache

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

Berechne die Grenzen:

1 - \frac{1}{e}

= 1 - \frac{1}{e}

\int_{1}^{2} \frac{1}{x ^{2} + 2 x + 5} d x

\int_{0}^{10} 2 e^{- 2 x} d x

\int_{0}^{2} 20 t^{2} e^{- t} d t

\int_{0}^{1} y e^{- 2 y^{2}} d y

\int_{0}^{a} cosh^{2} (\frac{t}{a}) d t