integral from 0 to sqrt(2 of)x^5*e^{x^2}

Lösung

\int_{0}^{2} x^{5} \cdot e^{x^{2}} d x

3 e^{4^{\frac{2}{5}} 52} - 4^{\frac{2}{5}} 52 e^{4^{\frac{2}{5}} 52} - 1

Dezimale

6.38905 \dots

Schritte zur Lösung

\int_{0}^{2} x^{5} e^{x^{2}} d x

Wende U-Substitution an

= \int_{0}^{42} \frac{e ^{u^{\frac{2}{5}}} u ^{\frac{1}{5}}}{5} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int_{0}^{42} e^{u^{\frac{2}{5}}} u^{\frac{1}{5}} d u

Wende U-Substitution an

= \frac{1}{5} \cdot \int_{0}^{4^{\frac{2}{5}} 52} \frac{5 e ^{v} v ^{2}}{2} d v

Entferne die Konstante:

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

= \frac{1}{5} \cdot \frac{5}{2} \cdot \int_{0}^{4^{\frac{2}{5}} 52} e^{v} v^{2} d v

Wende die partielle Integration an

= \frac{1}{5} \cdot \frac{5}{2} [v^{2} e^{v} - \int 2 v e^{v} d v]_{0}^{4^{\frac{2}{5}} 52}

\int 2 v e^{v} d v = 2 (e^{v} v - e^{v})

= \frac{1}{5} \cdot \frac{5}{2} [v^{2} e^{v} - 2 (e^{v} v - e^{v})]_{0}^{4^{\frac{2}{5}} 52}

Vereinfache

\frac{1}{5} \cdot \frac{5}{2} [v^{2} e^{v} - 2 (e^{v} v - e^{v})]_{0}^{4^{\frac{2}{5}} 52} : \frac{1}{2} [e^{v} v^{2} - 2 (e^{v} v - e^{v})]_{0}^{4^{\frac{2}{5}} 52}

= \frac{1}{2} [e^{v} v^{2} - 2 (e^{v} v - e^{v})]_{0}^{4^{\frac{2}{5}} 52}

Berechne die Grenzen:

6 e^{4^{\frac{2}{5}} 52} - 2 \cdot 4^{\frac{2}{5}} 52 e^{4^{\frac{2}{5}} 52} - 2

= \frac{1}{2} (6 e^{4^{\frac{2}{5}} 52} - 2 \cdot 4^{\frac{2}{5}} 52 e^{4^{\frac{2}{5}} 52} - 2)

Vereinfache

= 3 e^{4^{\frac{2}{5}} 52} - 4^{\frac{2}{5}} 52 e^{4^{\frac{2}{5}} 52} - 1

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