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

The integral of 2x^3e^{-x^2} is -e^{-x^2}x^2-e^{-x^2}+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \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

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

Apply Integration By Parts

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

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

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

Substitute back

u = - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of 2x^3e^{-x^2} ?
The integral of 2x^3e^{-x^2} is -e^{-x^2}x^2-e^{-x^2}+C