integral from-1 to 2 of x^3e^{x^2}

Lösung

\int_{- 1}^{2} x^{3} e^{x^{2}} d x

\frac{3 e ^{4}}{2}

Dezimale

81.89722 \dots

Schritte zur Lösung

\int_{- 1}^{2} x^{3} e^{x^{2}} d x

Wende U-Substitution an

= \int_{- 1}^{8} \frac{e ^{u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int_{- 1}^{8} e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Wende U-Substitution an

= \frac{1}{3} \cdot \int_{(- 1)^{\frac{2}{3}}}^{4} \frac{3 e ^{v} v}{2} d v

Entferne die Konstante:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int_{(- 1)^{\frac{2}{3}}}^{4} e^{v} v d v

Wende die partielle Integration an

= \frac{1}{3} \cdot \frac{3}{2} [e^{v} v - \int e^{v} d v]_{(- 1)^{\frac{2}{3}}}^{4}

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

= \frac{1}{3} \cdot \frac{3}{2} [e^{v} v - e^{v}]_{(- 1)^{\frac{2}{3}}}^{4}

Vereinfache

\frac{1}{3} \cdot \frac{3}{2} [e^{v} v - e^{v}]_{(- 1)^{\frac{2}{3}}}^{4} : \frac{1}{2} [e^{v} v - e^{v}]_{1}^{4}

= \frac{1}{2} [e^{v} v - e^{v}]_{1}^{4}

Berechne die Grenzen:

3 e^{4}

= \frac{1}{2} \cdot 3 e^{4}

Vereinfache

= \frac{3 e ^{4}}{2}

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