integral from 0 to x of (1-t)/(e^t)

Lösung

\int_{0}^{x} \frac{1 - t}{e ^{t}} d t

\frac{x}{e ^{x}}

Schritte zur Lösung

\int_{0}^{x} \frac{1 - t}{e ^{t}} d t

Wende U-Substitution an

= \int_{1}^{e^{x}} \frac{1 - ln ( u )}{u ^{2}} d u

Wende die partielle Integration an

= [- \frac{1 - ln ( u )}{u} - \int \frac{1}{u ^{2}} d u]_{1}^{e^{x}}

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= [- \frac{1 - ln ( u )}{u} - (- \frac{1}{u})]_{1}^{e^{x}}

Vereinfache

[- \frac{1 - ln ( u )}{u} - (- \frac{1}{u})]_{1}^{e^{x}} : [\frac{ln ( u )}{u}]_{1}^{e^{x}}

= [\frac{ln ( u )}{u}]_{1}^{e^{x}}

Berechne die Grenzen:

\frac{x}{e ^{x}}

= \frac{x}{e ^{x}}