de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (8-e^x)/(e^{5x)}

Lösung

\int \frac{8 - e ^{x}}{e ^{5 x}} d x

Lösung

- \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

Schritte zur Lösung

\int \frac{8 - e ^{x}}{e ^{5 x}} d x

Wende U-Substitution an

= \int \frac{8 - u}{u ^{6}} d u

Multipliziere aus

\frac{8 - u}{u ^{6}} : \frac{8}{u ^{6}} - \frac{1}{u ^{5}}

= \int \frac{8}{u ^{6}} - \frac{1}{u ^{5}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{8}{u ^{6}} d u - \int \frac{1}{u ^{5}} d u

\int \frac{8}{u ^{6}} d u = - \frac{8}{5 u ^{5}}

\int \frac{1}{u ^{5}} d u = - \frac{1}{4 u ^{4}}

= - \frac{8}{5 u ^{5}} - (- \frac{1}{4 u ^{4}})

Setze in

u = e^{x}

ein

= - \frac{8}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}})

Vereinfache

- \frac{8}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}}) : - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

= - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

Füge eine Konstante zur Lösung hinzu

= - \frac{8}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

Graph

Beliebte Beispiele

derivative 3^xtan(ln(x))

d er i v a t i v e 3^{x} tan (ln (x))

integral of x/(x^4)

\int \frac{x}{x ^{4}} d x

limit as x approaches 0 of (tan(x))/(3x)

x \to 0 lim (\frac{tan ( x )}{3 x})

ydy=4x(y^2+1)^{1/2}dx

y d y = 4 x (y^{2} + 1)^{\frac{1}{2}} d x

y^{''}+25y=-40sin(5t)

y^{^{''}} + 25 y = - 40 sin (5 t)