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

The integral of e^{(-x)/k}*(-x)/k is e^{-x/k}(x+k)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{(-x)/k}*(-x)/k

\int e^{\frac{- x}{k}} \cdot \frac{- x}{k} d x

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

Solution steps

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

Take the constant out:

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

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

Simplify

= (- \frac{1}{k}) \cdot \int x e^{\frac{- x}{k}} d x

Simplify

= (- \frac{1}{k}) \cdot \int e^{- \frac{x}{k}} x d x

Apply Integration By Parts

: - k e^{- \frac{x}{k}} x - \int - k e^{- \frac{x}{k}} d x

= (- \frac{1}{k}) (- k e^{- \frac{x}{k}} x - \int - k e^{- \frac{x}{k}} d x)

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

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

Simplify

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

= e^{- \frac{x}{k}} (x + k)

Add a constant to the solution

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

x \to \infty lim (x^{3} - 2 x)

f (x) = \frac{2 x + 5}{x + 2} \cdot 2 x - 5

\frac{d y}{d x} = x y + y^{2}

\int \frac{12 x ^{2}}{4 x ^{3} - 6} d x

y^{^{'}} = \frac{( 3 x sec ( \frac{y}{x} ) + y )}{x}

What is the integral of e^{(-x)/k}*(-x)/k ?
The integral of e^{(-x)/k}*(-x)/k is e^{-x/k}(x+k)+C