What is the integral of (4x^3)/(e^{2x^2)} ?

The integral of (4x^3)/(e^{2x^2)} is 1/2 (-2e^{-2x^2}x^2-e^{-2x^2})+C

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integral of (4x^3)/(e^{2x^2)}

\int \frac{4 x ^{3}}{e ^{2 x^{2}}} d x

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

Solution steps

\int \frac{4 x ^{3}}{e ^{2 x^{2}}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x ^{3}}{e ^{2 x^{2}}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

= 4 \cdot \int x^{3} e^{- 2 x^{2}} d x

Apply u-substitution

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

= 4 \cdot \int \frac{e ^{- 2 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

= 4 \cdot \frac{1}{3} \cdot \int e^{- 2 u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Apply u-substitution

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

= 4 \cdot \frac{1}{3} \cdot \int \frac{3 e ^{- 2 v} v}{2} d v

Take the constant out:

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

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{- 2 v} v d v

Apply u-substitution

: \int \frac{e ^{w} w}{4} d w

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int \frac{e ^{w} w}{4} d w

Take the constant out:

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

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{4} \cdot \int e^{w} w d w

Apply Integration By Parts

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

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{4} (e^{w} w - \int e^{w} d w)

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

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{4} (e^{w} w - e^{w})

Substitute back

= 4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{4} (e^{- 2 (x^{3})^{\frac{2}{3}}} (- 2 (x^{3})^{\frac{2}{3}}) - e^{- 2 (x^{3})^{\frac{2}{3}}})

Simplify

4 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \frac{1}{4} (e^{- 2 (x^{3})^{\frac{2}{3}}} (- 2 (x^{3})^{\frac{2}{3}}) - e^{- 2 (x^{3})^{\frac{2}{3}}}) : \frac{1}{2} (- 2 e^{- 2 x^{2}} x^{2} - e^{- 2 x^{2}})

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{1}{( x - 2 ) ^{2}})

f (x) = x^{2} + 2 x + 1, at x = - 3

(8 - 4) (89)

\frac{d}{d x} (x^{2} - sin (x))

x^{2} + y^{2} = 169, (5, - 12)

What is the integral of (4x^3)/(e^{2x^2)} ?
The integral of (4x^3)/(e^{2x^2)} is 1/2 (-2e^{-2x^2}x^2-e^{-2x^2})+C