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

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

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

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

x + e^{- x^{2}} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

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

= x - (- e^{- x^{2}})

Simplify

= x + e^{- x^{2}}

Add a constant to the solution

= x + e^{- x^{2}} + C

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