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

The integral of (e^{-x}(3-3xln(x)))/(2x) is 3/2 e^{-x}ln(x)+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}(3-3xln(x)))/(2x)

\int \frac{e ^{- x} ( 3 - 3 x ln ( x ) )}{2 x} d x

\frac{3}{2} e^{- x} ln (x) + C

Solution steps

\int \frac{e ^{- x} ( 3 - 3 x ln ( x ) )}{2 x} 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 \frac{e ^{- x} ( 3 - 3 x ln ( x ) )}{x} d x

Expand

\frac{e ^{- x} ( 3 - 3 x ln ( x ) )}{x} : \frac{3 e ^{- x}}{x} - 3 e^{- x} ln (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{3 e ^{- x}}{x} d x - \int 3 e^{- x} ln (x) d x)

\int \frac{3 e ^{- x}}{x} d x = 3 Ei (- x)

\int 3 e^{- x} ln (x) d x = 3 (- e^{- x} ln (x) + Ei (- x))

= \frac{1}{2} (3 Ei (- x) - 3 (- e^{- x} ln (x) + Ei (- x)))

Simplify

\frac{1}{2} (3 Ei (- x) - 3 (- e^{- x} ln (x) + Ei (- x))) : \frac{3}{2} e^{- x} ln (x)

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

Add a constant to the solution

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

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

\int 4 - ∣ x ∣ d x

\int_{- 1}^{1} x + 1 - 1 d x

\int_{- 3}^{2} ∣ x + 1 ∣ d x

x \to 1 lim (\frac{2 x}{ln ( x )})

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