integral from 0 to 8 of 2xe^{-0.2x}

Lösung

\int_{0}^{8} 2 x e^{- 0.2 x} d x

23.75345 \dots

Schritte zur Lösung

\int_{0}^{8} 2 x e^{- 0.2 x} d x

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{8} x e^{- 0.2 x} d x

Wende U-Substitution an

= 2 \cdot \int_{0}^{- 1.6} \frac{5}{0.2} e^{u} u d u

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

= 2 (- \int_{- 1.6}^{0} \frac{5}{0.2} e^{u} u d u)

Entferne die Konstante:

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

= 2 (- \frac{5}{0.2} \cdot \int_{- 1.6}^{0} e^{u} u d u)

Vereinfache

= 2 (- 25 \cdot \int_{- 1.6}^{0} e^{u} u d u)

Wende die partielle Integration an

= 2 (- 25 [e^{u} u - \int e^{u} d u]_{- 1.6}^{0})

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

= 2 (- 25 [e^{u} u - e^{u}]_{- 1.6}^{0})

Vereinfache

= - 50 [e^{u} u - e^{u}]_{- 1.6}^{0}

Berechne die Grenzen:

- 0.47506 \dots

= - 50 (- 0.47506 \dots)

Vereinfache

= 23.75345 \dots

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

\int_{0}^{7} \frac{1}{( x - 3 ) ^{2}} d x

\int_{1}^{2} x \frac{3}{7} x^{2} d x

\int_{0}^{\frac{π}{5}} x^{2} sin (3 x) d x

\int_{0}^{2} 9 x^{2} - e^{x} d x