What is the integral of x^5e^{-x^2} ?

The integral of x^5e^{-x^2} is -1/2 e^{-x^2}x^4-e^{-x^2}x^2-e^{-x^2}+C

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integral of x^5e^{-x^2}

\int x^{5} e^{- x^{2}} d x

- \frac{1}{2} e^{- x^{2}} x^{4} - e^{- x^{2}} x^{2} - e^{- x^{2}} + C

Solution steps

\int x^{5} e^{- x^{2}} d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x^{2}

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

Simplify

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

= - \frac{1}{2} e^{- x^{2}} x^{4} - e^{- x^{2}} x^{2} - e^{- x^{2}}

Add a constant to the solution

= - \frac{1}{2} e^{- x^{2}} x^{4} - e^{- x^{2}} x^{2} - e^{- x^{2}} + C

f (x) = \frac{2}{3} x^{3} + 4 x^{2} + 6 x - 5

x \to 4 + lim (x + 1)

\int x^{2} x d x

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

\frac{d y}{d t} = k y^{2}

What is the integral of x^5e^{-x^2} ?
The integral of x^5e^{-x^2} is -1/2 e^{-x^2}x^4-e^{-x^2}x^2-e^{-x^2}+C