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

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

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

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

\frac{π}{2 e} erf (x) + C

Solution steps

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

Simplify

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

Simplify

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

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

Take the constant out:

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

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

Apply exponent rule:

a^{- 1} = \frac{1}{a}

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

Use the nonelementary integral:

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

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

Simplify

\frac{1}{e} \cdot \frac{π}{2} erf (x) : \frac{π}{2 e} erf (x)

= \frac{π}{2 e} erf (x)

Add a constant to the solution

= \frac{π}{2 e} erf (x) + C

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