integral from 0 to sqrt(2 of)x^3e^{x^2}

Lösung

\int_{0}^{2} x^{3} e^{x^{2}} d x

\frac{2 ^{\frac{2}{3}} 3 2 e ^{2^{\frac{2}{3}} 32} - e ^{2^{\frac{2}{3}} 32} + 1}{2}

Dezimale

4.19452 \dots

Schritte zur Lösung

\int_{0}^{2} x^{3} e^{x^{2}} d x

Wende U-Substitution an

= \int_{0}^{22} \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_{0}^{22} e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Wende U-Substitution an

= \frac{1}{3} \cdot \int_{0}^{2^{\frac{2}{3}} 32} \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_{0}^{2^{\frac{2}{3}} 32} 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]_{0}^{2^{\frac{2}{3}} 32}

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

= \frac{1}{3} \cdot \frac{3}{2} [e^{v} v - e^{v}]_{0}^{2^{\frac{2}{3}} 32}

Vereinfache

\frac{1}{3} \cdot \frac{3}{2} [e^{v} v - e^{v}]_{0}^{2^{\frac{2}{3}} 32} : \frac{1}{2} [e^{v} v - e^{v}]_{0}^{2^{\frac{2}{3}} 32}

= \frac{1}{2} [e^{v} v - e^{v}]_{0}^{2^{\frac{2}{3}} 32}

Berechne die Grenzen:

2^{\frac{2}{3}} 32 e^{2^{\frac{2}{3}} 32} - e^{2^{\frac{2}{3}} 32} + 1

= \frac{1}{2} (2^{\frac{2}{3}} 32 e^{2^{\frac{2}{3}} 32} - e^{2^{\frac{2}{3}} 32} + 1)

Vereinfache

= \frac{2 ^{\frac{2}{3}} 3 2 e ^{2^{\frac{2}{3}} 32} - e ^{2^{\frac{2}{3}} 32} + 1}{2}

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