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

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

integral

\frac{x}{( e ^{x} ) ^{2}}

- \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x} + C

Solution steps

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

(e^{x})^{2} = e^{2 x}

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

Apply Integration By Parts

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

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

\int - \frac{1}{2} e^{- 2 x} d x = \frac{1}{4} e^{- 2 x}

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

Add a constant to the solution

= - \frac{1}{2} e^{- 2 x} x - \frac{1}{4} e^{- 2 x} + C

(e^{x} + 2)^{2}

\frac{2 - x}{1 - x}

\frac{tan ( x )}{( 1 + cos ^{4} ( x ) )}

2 e^{- 2 x}

\frac{1}{( x )}

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