What is the integral of xe^{-x^4} ?

The integral of xe^{-x^4} is (sqrt(pi))/4 \erf(sqrt(x^4))+C

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integral of xe^{-x^4}

\int x e^{- x^{4}} d x

\frac{π}{4} erf (x^{4}) + C

Solution steps

\int x e^{- x^{4}} d x

Apply u-substitution

: \int \frac{e ^{- u}}{4 u} d u

= \int \frac{e ^{- u}}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{e ^{- u}}{u} d u

Apply u-substitution

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

= \frac{1}{4} \cdot \int 2 e^{- v^{2}} d v

Take the constant out:

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

= \frac{1}{4} \cdot 2 \cdot \int e^{- v^{2}} d v

Use the nonelementary integral:

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

= \frac{1}{4} \cdot 2 \cdot \frac{π}{2} erf (v)

Substitute back

= \frac{1}{4} \cdot 2 \cdot \frac{π}{2} erf (x^{4})

Simplify

\frac{1}{4} \cdot 2 \cdot \frac{π}{2} erf (x^{4}) : \frac{π}{4} erf (x^{4})

= \frac{π}{4} erf (x^{4})

Add a constant to the solution

= \frac{π}{4} erf (x^{4}) + C

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What is the integral of xe^{-x^4} ?
The integral of xe^{-x^4} is (sqrt(pi))/4 \erf(sqrt(x^4))+C