What is the integral of (3x)/((x+2)(x-4)) ?

The integral of (3x)/((x+2)(x-4)) is ln|x+2|+2ln|x-4|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3x)/((x+2)(x-4))

\int \frac{3 x}{( x + 2 ) ( x - 4 )} d x

ln ∣ x + 2 ∣ + 2 ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{3 x}{( x + 2 ) ( x - 4 )} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{( x + 2 ) ( x - 4 )} d x

Take the partial fraction of

\frac{x}{( x + 2 ) ( x - 4 )} : \frac{1}{3 ( x + 2 )} + \frac{2}{3 ( x - 4 )}

= 3 \cdot \int \frac{1}{3 ( x + 2 )} + \frac{2}{3 ( x - 4 )} 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

= 3 (\int \frac{1}{3 ( x + 2 )} d x + \int \frac{2}{3 ( x - 4 )} d x)

\int \frac{1}{3 ( x + 2 )} d x = \frac{1}{3} ln ∣ x + 2 ∣

\int \frac{2}{3 ( x - 4 )} d x = \frac{2}{3} ln ∣ x - 4 ∣

= 3 (\frac{1}{3} ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 4 ∣)

Simplify

3 (\frac{1}{3} ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 4 ∣) : ln ∣ x + 2 ∣ + 2 ln ∣ x - 4 ∣

= ln ∣ x + 2 ∣ + 2 ln ∣ x - 4 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + 2 ln ∣ x - 4 ∣ + C

\int \frac{2 ^{x - 1} - 5 ^{x - 1}}{1 0 ^{x}} d x

\int c e^{- θ y} d y

\int 5 x sin (5 x) d x

\int 7 x^{3} d x

\int \frac{1 - 2 x}{x ^{2} + 9 x + 14} d x

What is the integral of (3x)/((x+2)(x-4)) ?
The integral of (3x)/((x+2)(x-4)) is ln|x+2|+2ln|x-4|+C