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

Lösung

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

\frac{1}{2} e^{x^{2} - 1} (x^{2} - 1) + C

Schritte zur Lösung

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

Vereinfache

e^{x^{2} - 1} : e^{x^{2}} e^{- 1}

= \int x^{3} e^{x^{2}} e^{- 1} d x

Entferne die Konstante:

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

= e^{- 1} \cdot \int x^{3} e^{x^{2}} d x

Wende Exponentenregel an:

a^{- 1} = \frac{1}{a}

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Wende die partielle Integration an

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

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

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t y = x, (16, 4)

y^{^{'}} + \frac{y}{x} = \frac{7}{x ^{2}} + 3

\int (e x^{2} + π) d x

x \to 0 lim (\frac{( 2 ∣ x ∣ )}{3 x})

\frac{\partial}{\partial x} (x^{2} + y^{2} - y cos (x))