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

Lösung

\int_{1}^{l n (2)} e^{- x} d x

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

Dezimale

- 0.13212 \dots

Schritte zur Lösung

\int_{1}^{l n (2)} e^{- x} d x

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

= - \int_{l n (2)}^{1} e^{- x} d x

Wende U-Substitution an

= - \int_{- l n (2)}^{- 1} - e^{u} d u

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

= - (- \int_{- 1}^{- l n (2)} - e^{u} d u)

Entferne die Konstante:

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

= - (- (- \int_{- 1}^{- l n (2)} e^{u} d u))

Nutze das gemeinsame Integral :

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

= - (- (- [e^{u}]_{- 1}^{- l n (2)}))

Vereinfache

= - [e^{u}]_{- 1}^{- l n (2)}

Berechne die Grenzen:

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

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

Vereinfache

= - \frac{1}{2} + \frac{1}{e}

\int_{- 2}^{3} \frac{1}{( x + 1 ) ^{3}} d x

\int_{0}^{1} (e^{x}) - (e^{- x}) d x

\int_{0}^{7} e^{x - 7} d x

\int_{x}^{y} e^{- t^{2}} d t

\int_{0}^{1} x^{2} + \frac{x y}{3} d x