What is the integral of x^5e^{-x^4} ?

The integral of x^5e^{-x^4} is 1/2 (-1/2 e^{-x^4}x^2+(sqrt(pi))/4 \erf(x^2))+C

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integral of x^5e^{-x^4}

\int x^{5} e^{- x^{4}} d x

\frac{1}{2} (- \frac{1}{2} e^{- x^{4}} x^{2} + \frac{π}{4} erf (x^{2})) + C

Solution steps

\int x^{5} e^{- x^{4}} d x

Apply u-substitution

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

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

Simplify

u^{2} : uu

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

Apply Integration By Parts

: - \frac{1}{2} e^{- u^{2}} u - \int - \frac{1}{2} e^{- u^{2}} d u

= \frac{1}{2} (- \frac{1}{2} e^{- u^{2}} u - \int - \frac{1}{2} e^{- u^{2}} d u)

\int - \frac{1}{2} e^{- u^{2}} d u = - \frac{π}{4} erf (u)

= \frac{1}{2} (- \frac{1}{2} e^{- u^{2}} u - (- \frac{π}{4} erf (u)))

Substitute back

u = x^{2}

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

Simplify

\frac{1}{2} (- \frac{1}{2} e^{- (x^{2})^{2}} x^{2} - (- \frac{π}{4} erf (x^{2}))) : \frac{1}{2} (- \frac{1}{2} e^{- x^{4}} x^{2} + \frac{π}{4} erf (x^{2}))

= \frac{1}{2} (- \frac{1}{2} e^{- x^{4}} x^{2} + \frac{π}{4} erf (x^{2}))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{2} e^{- x^{4}} x^{2} + \frac{π}{4} erf (x^{2})) + C

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What is the integral of x^5e^{-x^4} ?
The integral of x^5e^{-x^4} is 1/2 (-1/2 e^{-x^4}x^2+(sqrt(pi))/4 \erf(x^2))+C