What is the integral from x to sqrt(x of)e^{x/y} ?

The integral from x to sqrt(x of)e^{x/y} is -x(-(e^{sqrt(x)})/(sqrt(x))+\Ei(sqrt(x))+e-\Ei(1))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

\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

Apply Integration By Parts

= - 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}}}

Simplify

- 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}

Compute the boundaries:

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

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

What is the integral from x to sqrt(x of)e^{x/y} ?
The integral from x to sqrt(x of)e^{x/y} is -x(-(e^{sqrt(x)})/(sqrt(x))+\Ei(sqrt(x))+e-\Ei(1))