What is the integral of ye^{y/x} ?

The integral of ye^{y/x} is -y^2(-(e^{y/x}x)/y+\Ei(y/x))+C

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integral of ye^{y/x}

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

- y^{2} (- \frac{e ^{\frac{y}{x}} x}{y} + Ei (\frac{y}{x})) + C

Solution steps

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

Take the constant out:

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

= y \cdot \int e^{\frac{y}{x}} d x

Apply u-substitution

: \int - \frac{y e ^{u}}{u ^{2}} d u

= y \cdot \int - \frac{y e ^{u}}{u ^{2}} d u

Take the constant out:

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

= y (- y \cdot \int \frac{e ^{u}}{u ^{2}} d u)

Apply Integration By Parts

: - \frac{e ^{u}}{u} - \int - \frac{e ^{u}}{u} d u

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

Take the constant out:

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

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

Use the nonelementary integral:

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

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

Substitute back

u = \frac{y}{x}

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

Simplify

y (- y (- \frac{e ^{\frac{y}{x}}}{\frac{y}{x}} - (- Ei (\frac{y}{x})))) : - y^{2} (- \frac{e ^{\frac{y}{x}} x}{y} + Ei (\frac{y}{x}))

= - y^{2} (- \frac{e ^{\frac{y}{x}} x}{y} + Ei (\frac{y}{x}))

Add a constant to the solution

= - y^{2} (- \frac{e ^{\frac{y}{x}} x}{y} + Ei (\frac{y}{x})) + C

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What is the integral of ye^{y/x} ?
The integral of ye^{y/x} is -y^2(-(e^{y/x}x)/y+\Ei(y/x))+C