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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2/(x(x+2))

\int \frac{2}{x ( x + 2 )} d x

ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2}{x ( x + 2 )} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ( x + 2 )} d x

Take the partial fraction of

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

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

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

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

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

Simplify

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

= ln ∣ x ∣ - ln ∣ x + 2 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

\int \frac{4 x + 6}{2 x + 1} d x

\int 5 cot^{2} (x) csc^{4} (x) d x

\int y^{6} cos (3 y) d y

\int sin (3 x) cos (8 x) d x

\int t^{\frac{1}{4}} d t

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