integral from x to sqrt(x of)e^{x/y}

Lösung

\int_{x}^{x} e^{\frac{x}{y}} d y

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

Schritte zur Lösung

\int_{x}^{x} e^{\frac{x}{y}} d y

Wende U-Substitution an

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

Entferne die Konstante:

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

= - x \cdot \int_{1}^{x^{\frac{1}{2}}} \frac{e ^{u}}{u ^{2}} d u

Wende die partielle Integration an

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

\int - \frac{e ^{u}}{u} d u = - Ei (u)

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

Vereinfache

- x [- \frac{e ^{u}}{u} - (- Ei (u))]_{1}^{x^{\frac{1}{2}}} : - x [- \frac{e ^{u}}{u} + Ei (u)]_{1}^{x}

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

Berechne die Grenzen:

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

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

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