What is the integral of y^{-1}e^{-y^2} ?

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

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integral of y^{-1}e^{-y^2}

\int y^{- 1} e^{- y^{2}} d y

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

Solution steps

\int y^{- 1} e^{- y^{2}} d y

Simplify

y^{- 1} e^{- y^{2}} : \frac{e ^{- y^{2}}}{y}

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

Apply u-substitution

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

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

Apply u-substitution

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

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

Use the nonelementary integral:

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

= \frac{1}{2} Ei (v)

Substitute back

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

Add a constant to the solution

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

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