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

The integral of (2x+2)/(2x^2+x) is -ln|2x+1|+2ln|x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int \frac{2 x + 2}{2 x ^{2} + x} d x

- ln ∣ 2 x + 1 ∣ + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{2 x + 2}{2 x ^{2} + x} d x

Expand

\frac{2 x + 2}{2 x ^{2} + x} : \frac{2 x}{2 x ^{2} + x} + \frac{2}{2 x ^{2} + x}

Apply the Sum Rule:

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

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

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

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

= ln ∣ 2 x + 1 ∣ + 2 (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣)

Simplify

ln ∣ 2 x + 1 ∣ + 2 (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣) : - ln ∣ 2 x + 1 ∣ + 2 ln ∣ x ∣

= - ln ∣ 2 x + 1 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= - ln ∣ 2 x + 1 ∣ + 2 ln ∣ x ∣ + C

\int \frac{4}{x x ^{2} - 36} d x

\int \frac{1}{y I n ^{2} y} d y

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

\int \frac{x ^{2}}{10 x - x ^{2}} d x

\int 5 x^{2} + x d x

What is the integral of (2x+2)/(2x^2+x) ?
The integral of (2x+2)/(2x^2+x) is -ln|2x+1|+2ln|x|+C