What is the integral of (x+4)/(x^3+3x^2+2x) ?

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

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integral of (x+4)/(x^3+3x^2+2x)

\int \frac{x + 4}{x ^{3} + 3 x ^{2} + 2 x} d x

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

Solution steps

\int \frac{x + 4}{x ^{3} + 3 x ^{2} + 2 x} d x

Take the partial fraction of

\frac{x + 4}{x ^{3} + 3 x ^{2} + 2 x} : \frac{2}{x} - \frac{3}{x + 1} + \frac{1}{x + 2}

= \int \frac{2}{x} - \frac{3}{x + 1} + \frac{1}{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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

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

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

Add a constant to the solution

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

\frac{d}{d x} ((a^{\frac{2}{3}} - x^{\frac{2}{3}})^{\frac{3}{2}})

\frac{d}{d x} (\frac{arcsin ( x )}{arctan ( x )})

f (x) = x^{3} - 4 x

\int (tan (x) sec^{2} (y)) d x

\frac{d}{d x} ((a + b x)^{3})

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