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

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

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

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

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

Solution steps

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

Simplify

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

Take the constant out:

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

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

\int_{0}^{1} 2 - x^{2} d x

x \to 0 lim (\frac{1 + cos ( x )}{x})

\int \frac{x}{( x ^{2} + 1 ) ^{(\frac{3}{2})}} d x

x y^{^{'}} = 3 (x + y)

d er i v a t i v e 1.0 3^{x}

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