What is the integral of xe^{1/x} ?

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

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integral of xe^{1/x}

\int x e^{\frac{1}{x}} d x

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

Solution steps

\int x e^{\frac{1}{x}} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

x \to - 2 lim (\frac{x ^{2} - 4}{x + 2})

e^{\frac{6}{x}}

x \to 3.01 lim (x^{2} + 3 x)

x \frac{d y}{d x} = 2 x e^{x^{2}} - y + 6 x^{2}

y^{^{'}} + y - 3 cos (x) = 0

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