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

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

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integral of xe^{x^{-2}}

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

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

Solution steps

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

x^{- 2} = \frac{1}{x ^{2}}

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

Apply u-substitution

: \int \frac{e ^{\frac{1}{u}}}{2} d u

= \int \frac{e ^{\frac{1}{u}}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{\frac{1}{u}} d u

Apply u-substitution

: \int - \frac{e ^{v}}{v ^{2}} d v

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Use the nonelementary integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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