integral from 0 to 3 of 3e^{-x}

Lösung

\int_{0}^{3} 3 e^{- x} d x

3 (1 - \frac{1}{e ^{3}})

Dezimale

2.85063 \dots

Schritte zur Lösung

\int_{0}^{3} 3 e^{- x} d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{3} e^{- x} d x

Wende U-Substitution an

= 3 \cdot \int_{0}^{- 3} - e^{u} d u

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

= 3 (- \int_{- 3}^{0} - e^{u} d u)

Entferne die Konstante:

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

= 3 (- (- \int_{- 3}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 3 (- (- [e^{u}]_{- 3}^{0}))

Vereinfache

= 3 [e^{u}]_{- 3}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{3}}

= 3 (1 - \frac{1}{e ^{3}})

\int_{1}^{32} \frac{6}{5} x^{\frac{1}{5}} d x

\int_{0}^{4} (x^{2} - x) d x

\int_{0}^{2 π} 60 sin (2 θ) d θ

\int_{1}^{3} (\frac{9}{x} - e^{- x}) d x

\int_{- 2}^{2} \frac{1}{4} e^{t x} d x