What is the integral of xe^{2/x} ?

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

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integral of xe^{2/x}

\int x e^{\frac{2}{x}} d x

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

Solution steps

\int x e^{\frac{2}{x}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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