integral of (3-e^x)/(e^{5x)}

Lösung

\int \frac{3 - e ^{x}}{e ^{5 x}} d x

- \frac{3}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

Schritte zur Lösung

\int \frac{3 - e ^{x}}{e ^{5 x}} d x

Wende U-Substitution an

= \int \frac{3 - u}{u ^{6}} d u

Multipliziere aus

\frac{3 - u}{u ^{6}} : \frac{3}{u ^{6}} - \frac{1}{u ^{5}}

= \int \frac{3}{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{3}{u ^{6}} d u - \int \frac{1}{u ^{5}} d u

\int \frac{3}{u ^{6}} d u = - \frac{3}{5 u ^{5}}

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

= - \frac{3}{5 u ^{5}} - (- \frac{1}{4 u ^{4}})

Setze in

u = e^{x}

ein

= - \frac{3}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}})

Vereinfache

- \frac{3}{5 ( e ^{x} ) ^{5}} - (- \frac{1}{4 ( e ^{x} ) ^{4}}) : - \frac{3}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

= - \frac{3}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}}

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{5 e ^{5 x}} + \frac{1}{4 e ^{4 x}} + C

\int 3 - \frac{4}{2 x + 1} d x

f (x) = 3 x^{5} - 4 x^{4} + 9 x - 6

\int_{- 2}^{2} 2 - x^{2} d x

t an g e n t (x^{2} + 1)^{x + 1}

\frac{d}{d x} (x^{3} + 4 x - 15)