What is the integral of 4x^3e^{-2x^2} ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 4x^3e^{-2x^2}

\int 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 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 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

\int cot^{6} (3 x) d x

\int x^{3} x + 3 d x

\int (x^{2} + 1) e^{- x + 2} d x

\int \frac{3 x + 2}{x - 3} d x

\int \frac{x + 2}{3 x} d x

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