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

The integral of (x+3)/(x^2+2x) is -1/2 ln|x+2|+3/2 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 (x+3)/(x^2+2x)

\int \frac{x + 3}{x ^{2} + 2 x} d x

- \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{2} ln ∣ x ∣ + C

Solution steps

\int \frac{x + 3}{x ^{2} + 2 x} d x

Expand

\frac{x + 3}{x ^{2} + 2 x} : \frac{x}{x ^{2} + 2 x} + \frac{3}{x ^{2} + 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{x}{x ^{2} + 2 x} d x + \int \frac{3}{x ^{2} + 2 x} d x

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

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

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{2} ln ∣ x ∣ + C

\int \frac{1 - sin ( x )}{x} d x

\int \frac{5 x - 5}{x ^{2} - x - 6} d x

\int 2^{\frac{1}{x}} d x

\int x^{7} ln (9 x^{4}) d x

\int \frac{2 - x}{3 x ^{2} + 3 x} d x

What is the integral of (x+3)/(x^2+2x) ?
The integral of (x+3)/(x^2+2x) is -1/2 ln|x+2|+3/2 ln|x|+C