integral of (e^{\sqrt[5]{t}})/(sqrt(t))

Lösung

\int \frac{e ^{5 t}}{t} d t

5 (e^{5 t} t^{\frac{3}{10}} - \frac{3}{2} (e^{5 t} 10 t - \frac{π}{2} erfi (5 t))) + C

Schritte zur Lösung

\int \frac{e ^{5 t}}{t} d t

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

= 5 (e^{u} u^{\frac{3}{2}} - \int \frac{3}{2} e^{u} u d u)

\int \frac{3}{2} e^{u} u d u = \frac{3}{2} (e^{u} u - \frac{π}{2} erfi (u))

= 5 (e^{u} u^{\frac{3}{2}} - \frac{3}{2} (e^{u} u - \frac{π}{2} erfi (u)))

Setze in

u = 5 t

ein

= 5 (e^{5 t} (5 t)^{\frac{3}{2}} - \frac{3}{2} (e^{5 t} 5 t - \frac{π}{2} erfi (5 t)))

Vereinfache

5 (e^{5 t} (5 t)^{\frac{3}{2}} - \frac{3}{2} (e^{5 t} 5 t - \frac{π}{2} erfi (5 t))) : 5 (e^{5 t} t^{\frac{3}{10}} - \frac{3}{2} (e^{5 t} 10 t - \frac{π}{2} erfi (5 t)))

= 5 (e^{5 t} t^{\frac{3}{10}} - \frac{3}{2} (e^{5 t} 10 t - \frac{π}{2} erfi (5 t)))

Füge eine Konstante zur Lösung hinzu

= 5 (e^{5 t} t^{\frac{3}{10}} - \frac{3}{2} (e^{5 t} 10 t - \frac{π}{2} erfi (5 t))) + C

x \to 0 - lim (\frac{x + 3}{x})

\frac{d}{d x} (\frac{x ^{4} + 2 x ^{2}}{x ^{2}})

d er i v a t i v e 3 t^{2} - 12

a re a y = x^{2}, y = 4 x - x^{2}

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