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integral of x^3e^{x^2+1}

Lösung

\int x^{3} e^{x^{2} + 1} d x

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

Schritte zur Lösung

\int x^{3} e^{x^{2} + 1} d x

Wende U-Substitution an

= \int \frac{e ^{u^{\frac{2}{3}} + 1} 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 e^{u^{\frac{2}{3}} + 1} u^{\frac{1}{3}} d u

Vereinfache

e^{u^{\frac{2}{3}} + 1} : e^{u^{\frac{2}{3}}} e

= \frac{1}{3} \cdot \int e^{u^{\frac{2}{3}}} e u^{\frac{1}{3}} d u

Entferne die Konstante:

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

= \frac{1}{3} e \cdot \int e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Wende U-Substitution an

= \frac{1}{3} e \cdot \int \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} e \frac{3}{2} \cdot \int e^{v} v d v

Wende die partielle Integration an

= \frac{1}{3} e \frac{3}{2} (e^{v} v - \int e^{v} d v)

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

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

Ersetze zurück

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

Vereinfache

\frac{1}{3} e \frac{3}{2} (e^{(x^{3})^{\frac{2}{3}}} (x^{3})^{\frac{2}{3}} - e^{(x^{3})^{\frac{2}{3}}}) : \frac{e}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (2 ln (x y))

\int - x^{0} d x

in v erse l a pl a ce \frac{0.2}{( 10 s ^{3} + s ^{2} )}

\int (tan (x) sec^{2} (x)) d x

l a pl a ce t r an s f or m sin (\frac{1}{2} t) \cdot t^{2}