What is the integral of 6x^3e^{-x^2} ?

The integral of 6x^3e^{-x^2} is 3(-e^{-x^2}x^2-e^{-x^2})+C

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integral of 6x^3e^{-x^2}

\int 6 x^{3} e^{- x^{2}} d x

3 (- e^{- x^{2}} x^{2} - e^{- x^{2}}) + C

Solution steps

\int 6 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

= 6 \cdot \int x^{3} e^{- x^{2}} d x

Apply u-substitution

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

= 6 \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

= 6 \cdot \frac{1}{2} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 6 \cdot \frac{1}{2} (e^{u} u - \int e^{u} d u)

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

= 6 \cdot \frac{1}{2} (e^{u} u - e^{u})

Substitute back

u = - x^{2}

= 6 \cdot \frac{1}{2} (e^{- x^{2}} (- x^{2}) - e^{- x^{2}})

Simplify

6 \cdot \frac{1}{2} (e^{- x^{2}} (- x^{2}) - e^{- x^{2}}) : 3 (- e^{- x^{2}} x^{2} - e^{- x^{2}})

= 3 (- e^{- x^{2}} x^{2} - e^{- x^{2}})

Add a constant to the solution

= 3 (- e^{- x^{2}} x^{2} - e^{- x^{2}}) + C

\frac{d}{d z} (cos (z))

x \to 0 lim (π)

y = ln (\frac{7 + e ^{x}}{7 - e ^{x}})

\int cos^{2} (d) x d x

y^{^{'}} + 0.121 y = - 3

What is the integral of 6x^3e^{-x^2} ?
The integral of 6x^3e^{-x^2} is 3(-e^{-x^2}x^2-e^{-x^2})+C