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

The integral of (x(e^{-2x})/2) is 1/8 (-2e^{-2x}x-e^{-2x})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x(e^{-2x})/2)

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

\frac{1}{8} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int x e^{- 2 x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = - 2 x

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

Simplify

\frac{1}{2} \cdot \frac{1}{4} (e^{- 2 x} (- 2 x) - e^{- 2 x}) : \frac{1}{8} (- 2 e^{- 2 x} x - e^{- 2 x})

= \frac{1}{8} (- 2 e^{- 2 x} x - e^{- 2 x})

Add a constant to the solution

= \frac{1}{8} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

y = sin (x) + tan (x)

cos ((π x)^{3})

\frac{d}{d x} (ln (3 x^{2} - 2 x + 1))

\int_{0}^{π} cos^{3} (x) d x

ln (r)

What is the integral of (x(e^{-2x})/2) ?
The integral of (x(e^{-2x})/2) is 1/8 (-2e^{-2x}x-e^{-2x})+C