integral from 0 to 1 of xe^{-4x^2}

Lösung

\int_{0}^{1} x e^{- 4 x^{2}} d x

\frac{e ^{4} - 1}{8 e ^{4}}

Dezimale

0.12271 \dots

Schritte zur Lösung

\int_{0}^{1} x e^{- 4 x^{2}} d x

Wende U-Substitution an

= \int_{0}^{- 4} - \frac{e ^{u}}{8} d u

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

= - \int_{- 4}^{0} - \frac{e ^{u}}{8} d u

Entferne die Konstante:

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

= - (- \frac{1}{8} \cdot \int_{- 4}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- \frac{1}{8} [e^{u}]_{- 4}^{0})

Vereinfache

= \frac{1}{8} [e^{u}]_{- 4}^{0}

Berechne die Grenzen:

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

= \frac{1}{8} (1 - \frac{1}{e ^{4}})

Vereinfache

= \frac{e ^{4} - 1}{8 e ^{4}}

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