integral from 0 to 10 of 2e^{-2x}

Lösung

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

1 - \frac{1}{e ^{20}}

Dezimale

0.99999 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= 2 \cdot \int_{0}^{- 20} - \frac{1}{2} e^{u} d u

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

= 2 (- \int_{- 20}^{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

= 2 (- (- \frac{1}{2} \cdot \int_{- 20}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 2 (- (- \frac{1}{2} [e^{u}]_{- 20}^{0}))

Vereinfache

2 (- (- \frac{1}{2} [e^{u}]_{- 20}^{0})) : [e^{u}]_{- 20}^{0}

= [e^{u}]_{- 20}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{20}}

= 1 - \frac{1}{e ^{20}}

\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

\int_{1}^{b} \frac{1}{x ^{p}} d x

\int_{0}^{\frac{π}{4}} e^{- t} cos (t) d t