What is the integral of (1-e^{-x^2})/(x^2) ?

The integral of (1-e^{-x^2})/(x^2) is -1/x+(e^{-x^2})/x+sqrt(pi)\erf(x)+C

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integral of (1-e^{-x^2})/(x^2)

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{e ^{- x^{2}}}{x ^{2}} d x = - \frac{e ^{- x^{2}}}{x} - π erf (x)

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

Simplify

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

Add a constant to the solution

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

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