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

The integral of (2x+3)/(x^2+3x+2) is -3ln|2x+4|+3ln|2x+2|+4ln|x+2|-2ln|x+1|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x+3)/(x^2+3x+2)

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

- 3 ln ∣ 2 x + 4 ∣ + 3 ln ∣ 2 x + 2 ∣ + 4 ln ∣ x + 2 ∣ - 2 ln ∣ x + 1 ∣ + C

Solution steps

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

Apply Integration By Parts

: (2 x + 3) (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) - \int 2 (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) d x

= (2 x + 3) (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) - \int 2 (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) d x

\int 2 (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) d x = 2 (- x ln ∣ 2 x + 4 ∣ + x ln ∣ 2 x + 2 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣)

= (2 x + 3) (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) - 2 (- x ln ∣ 2 x + 4 ∣ + x ln ∣ 2 x + 2 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣)

Simplify

(2 x + 3) (- ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣) - 2 (- x ln ∣ 2 x + 4 ∣ + x ln ∣ 2 x + 2 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 1 ∣) : - 3 ln ∣ 2 x + 4 ∣ + 3 ln ∣ 2 x + 2 ∣ + 4 ln ∣ x + 2 ∣ - 2 ln ∣ x + 1 ∣

= - 3 ln ∣ 2 x + 4 ∣ + 3 ln ∣ 2 x + 2 ∣ + 4 ln ∣ x + 2 ∣ - 2 ln ∣ x + 1 ∣

Add a constant to the solution

= - 3 ln ∣ 2 x + 4 ∣ + 3 ln ∣ 2 x + 2 ∣ + 4 ln ∣ x + 2 ∣ - 2 ln ∣ x + 1 ∣ + C

\int tan^{4} (x) sec^{3} (x) tan (x) sec (x) d x

\int (x sin (x) - cos (x)) d x

\int \frac{x ^{3} - x ^{2}}{x ^{2} - 7 x + 10} d x

\int e^{2 x} sin (x) cos (x) d x

\int \frac{cos ^{2} ( x )}{sin ^{5} ( x )} d x

What is the integral of (2x+3)/(x^2+3x+2) ?
The integral of (2x+3)/(x^2+3x+2) is -3ln|2x+4|+3ln|2x+2|+4ln|x+2|-2ln|x+1|+C