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

Vereinfache

= \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_{0}^{2} x 3 4 x d x

\int_{1}^{10} x - 1 d x

\int_{\frac{π}{6}}^{\frac{π}{3}} cot^{4} (x) d x

\int_{- 64}^{8} \frac{1}{x ^{\frac{1}{3}}} d x

\int_{2}^{5} 5 x^{2} + 13 x - 6 d x