integral of x^9e^{x^5}

Lösung

\int x^{9} e^{x^{5}} d x

\frac{1}{5} (e^{x^{5}} x^{5} - e^{x^{5}}) + C

Schritte zur Lösung

\int x^{9} e^{x^{5}} d x

Wende U-Substitution an

= \int \frac{e ^{u^{\frac{5}{9}}} u ^{\frac{1}{9}}}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int e^{u^{\frac{5}{9}}} u^{\frac{1}{9}} d u

Wende U-Substitution an

= \frac{1}{9} \cdot \int \frac{9 e ^{v} v}{5} d v

Entferne die Konstante:

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

= \frac{1}{9} \cdot \frac{9}{5} \cdot \int e^{v} v d v

Wende die partielle Integration an

= \frac{1}{9} \cdot \frac{9}{5} (e^{v} v - \int e^{v} d v)

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

= \frac{1}{9} \cdot \frac{9}{5} (e^{v} v - e^{v})

Ersetze zurück

= \frac{1}{9} \cdot \frac{9}{5} (e^{(x^{9})^{\frac{5}{9}}} (x^{9})^{\frac{5}{9}} - e^{(x^{9})^{\frac{5}{9}}})

Vereinfache

\frac{1}{9} \cdot \frac{9}{5} (e^{(x^{9})^{\frac{5}{9}}} (x^{9})^{\frac{5}{9}} - e^{(x^{9})^{\frac{5}{9}}}) : \frac{1}{5} (e^{x^{5}} x^{5} - e^{x^{5}})

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (e^{x^{5}} x^{5} - e^{x^{5}}) + C

\frac{d}{d x} (\frac{a + b x + c x ^{2}}{x})

x \to 2 + lim (4 - x^{2})

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

s l o p e (6.3) (9.6)

y^{^{''}} - 4 y^{^{'}} + 5 y = 0