What is the integral of (4w+6)/(w^3+6w^2+11w+6) ?

The integral of (4w+6)/(w^3+6w^2+11w+6) is ln|w+1|+2ln|w+2|-3ln|w+3|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4w+6)/(w^3+6w^2+11w+6)

\int \frac{4 w + 6}{w ^{3} + 6 w ^{2} + 11 w + 6} d w

ln ∣ w + 1 ∣ + 2 ln ∣ w + 2 ∣ - 3 ln ∣ w + 3 ∣ + C

Solution steps

\int \frac{4 w + 6}{w ^{3} + 6 w ^{2} + 11 w + 6} d w

Take the partial fraction of

\frac{4 w + 6}{w ^{3} + 6 w ^{2} + 11 w + 6} : \frac{1}{w + 1} + \frac{2}{w + 2} - \frac{3}{w + 3}

= \int \frac{1}{w + 1} + \frac{2}{w + 2} - \frac{3}{w + 3} d w

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{1}{w + 1} d w + \int \frac{2}{w + 2} d w - \int \frac{3}{w + 3} d w

\int \frac{1}{w + 1} d w = ln ∣ w + 1 ∣

\int \frac{2}{w + 2} d w = 2 ln ∣ w + 2 ∣

\int \frac{3}{w + 3} d w = 3 ln ∣ w + 3 ∣

= ln ∣ w + 1 ∣ + 2 ln ∣ w + 2 ∣ - 3 ln ∣ w + 3 ∣

Add a constant to the solution

= ln ∣ w + 1 ∣ + 2 ln ∣ w + 2 ∣ - 3 ln ∣ w + 3 ∣ + C

\int sec^{2} (t) + sec (t) tan (t) d t

\int \frac{1}{( x ^{2} 9 - x ^{2} )} d x

\int (10 x + 5) d x

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

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

What is the integral of (4w+6)/(w^3+6w^2+11w+6) ?
The integral of (4w+6)/(w^3+6w^2+11w+6) is ln|w+1|+2ln|w+2|-3ln|w+3|+C