integral from 0 to 4 of 1/5 e^{-x/5}

Lösung

\int_{0}^{4} \frac{1}{5} e^{- \frac{x}{5}} d x

1 - \frac{1}{e ^{\frac{4}{5}}}

Dezimale

0.55067 \dots

Schritte zur Lösung

\int_{0}^{4} \frac{1}{5} e^{- \frac{x}{5}} d x

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int_{0}^{4} e^{- \frac{x}{5}} d x

Wende U-Substitution an

= \frac{1}{5} \cdot \int_{0}^{- \frac{4}{5}} - 5 e^{u} d u

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

= \frac{1}{5} (- \int_{- \frac{4}{5}}^{0} - 5 e^{u} d u)

Entferne die Konstante:

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

= \frac{1}{5} (- (- 5 \cdot \int_{- \frac{4}{5}}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= \frac{1}{5} (- (- 5 [e^{u}]_{- \frac{4}{5}}^{0}))

Vereinfache

\frac{1}{5} (- (- 5 [e^{u}]_{- \frac{4}{5}}^{0})) : [e^{u}]_{- \frac{4}{5}}^{0}

= [e^{u}]_{- \frac{4}{5}}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{\frac{4}{5}}}

= 1 - \frac{1}{e ^{\frac{4}{5}}}

\int_{- 2}^{5} (x^{2} + 6 x + 8) d x

\int_{4}^{\infty} ln (x) d x

\int_{n}^{2 n} \frac{1}{x} d x

\int_{0}^{5} x^{2} + 3 x + 1 d x

\int_{1}^{2} (6 x^{2} - 6 x + 6) d x