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

The integral of-xe^{y/x} is -1/2 e^{y/x}x^2-(ye^{y/x}x)/2+y y/2 \Ei(y/x)+C

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integral of-xe^{y/x}

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

- \frac{1}{2} e^{\frac{y}{x}} x^{2} - \frac{y e ^{\frac{y}{x}} x}{2} + y \frac{y}{2} Ei (\frac{y}{x}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{1}{2} e^{\frac{y}{x}} x^{2} - \int - \frac{y e ^{\frac{y}{x}}}{2} d x

= - (\frac{1}{2} e^{\frac{y}{x}} x^{2} - \int - \frac{y e ^{\frac{y}{x}}}{2} d x)

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

= - (\frac{1}{2} e^{\frac{y}{x}} x^{2} - (- \frac{y e ^{\frac{y}{x}} x}{2} + y \frac{y}{2} Ei (\frac{y}{x})))

Simplify

= - \frac{1}{2} e^{\frac{y}{x}} x^{2} - \frac{y e ^{\frac{y}{x}} x}{2} + y \frac{y}{2} Ei (\frac{y}{x})

Add a constant to the solution

= - \frac{1}{2} e^{\frac{y}{x}} x^{2} - \frac{y e ^{\frac{y}{x}} x}{2} + y \frac{y}{2} Ei (\frac{y}{x}) + C

\frac{d}{d x} (x^{2} + 5 x + 4)

y = x^{2} - 4 x, (3, - 3)

\frac{d}{d x} (9.75 + 1.5 x^{3})

\int_{0}^{8} \frac{2 x - 1}{x + 1} d x

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

What is the integral of-xe^{y/x} ?
The integral of-xe^{y/x} is -1/2 e^{y/x}x^2-(ye^{y/x}x)/2+y y/2 \Ei(y/x)+C