integral of (e^{x^2})/(2x)*5x^4

Lösung

\int \frac{e ^{x^{2}}}{2 x} \cdot 5 x^{4} d x

\frac{5}{4} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

Schritte zur Lösung

\int \frac{e ^{x^{2}}}{2 x} \cdot 5 x^{4} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot 5 \cdot \int \frac{e ^{x^{2}}}{x} x^{4} d x

\frac{1}{2} \cdot 5 = \frac{5}{2}

= \frac{5}{2} \cdot \int \frac{e ^{x^{2}}}{x} x^{4} d x

Vereinfache

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

= \frac{5}{2} \cdot \int e^{x^{2}} x^{3} d x

Wende U-Substitution an

= \frac{5}{2} \cdot \int \frac{e ^{u} u}{2} d u

Entferne die Konstante:

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

= \frac{5}{2} \cdot \frac{1}{2} \cdot \int e^{u} u d u

Wende die partielle Integration an

= \frac{5}{2} \cdot \frac{1}{2} (e^{u} u - \int e^{u} d u)

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

= \frac{5}{2} \cdot \frac{1}{2} (e^{u} u - e^{u})

Setze in

u = x^{2}

ein

= \frac{5}{2} \cdot \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

Vereinfache

\frac{5}{2} \cdot \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) : \frac{5}{4} (e^{x^{2}} x^{2} - e^{x^{2}})

= \frac{5}{4} (e^{x^{2}} x^{2} - e^{x^{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{5}{4} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

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