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

The integral of x^3e^{-ax^2} is (-ae^{-ax^2}x^2-e^{-ax^2})/(2a^2)+C

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

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

\frac{- a e ^{- a x^{2}} x ^{2} - e ^{- a x^{2}}}{2 a ^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{- a u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

= \int \frac{e ^{- a u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{- a u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Apply u-substitution

: \int \frac{3 e ^{- a v} v}{2} d v

= \frac{1}{3} \cdot \int \frac{3 e ^{- a v} v}{2} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{- a v} v d v

Apply u-substitution

: \int \frac{e ^{w} w}{a ^{2}} d w

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int \frac{e ^{w} w}{a ^{2}} d w

Take the constant out:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{a ^{2}} \cdot \int e^{w} w d w

Apply Integration By Parts

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{a ^{2}} (e^{w} w - \int e^{w} d w)

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{a ^{2}} (e^{w} w - e^{w})

Substitute back

= \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{a ^{2}} (e^{- a (x^{3})^{\frac{2}{3}}} (- a (x^{3})^{\frac{2}{3}}) - e^{- a (x^{3})^{\frac{2}{3}}})

Simplify

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

= \frac{- a e ^{- a x^{2}} x ^{2} - e ^{- a x^{2}}}{2 a ^{2}}

Add a constant to the solution

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

\int 54

(x)^{5 x}

\frac{( x ^{2} + 1 ) d y}{d x} + 3 x (y - 1) = 0

\frac{d}{d x} (arcsin (\frac{x ^{2} + y ^{2}}{2}))

y = \frac{6 x}{x ^{2} + 1}, (1, 3)

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