What is the integral of x^3*e^{-x^2} ?

The integral of x^3*e^{-x^2} is 1/2 (-e^{-x^2}x^2-e^{-x^2})+C

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integral of x^3*e^{-x^2}

\int x^{3} \cdot e^{- x^{2}} d x

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

Solution steps

\int x^{3} e^{- x^{2}} d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

\int e^{u} d u = e^{u}

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

Substitute back

u = - x^{2}

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

Simplify

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

Add a constant to the solution

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

x \to \infty lim (arctan (ln (x)))

\frac{d ^{3}}{d x ^{3}} (\frac{6}{x})

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

4 x^{\frac{5}{4}} + 8 x^{\frac{3}{2}} + 10 x

\frac{\partial}{\partial x} (e^{t - x})

What is the integral of x^3*e^{-x^2} ?
The integral of x^3*e^{-x^2} is 1/2 (-e^{-x^2}x^2-e^{-x^2})+C