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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

Add a constant to the solution

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

x \to + 4 lim (3 e^{1.7 x} + 1)

x^{3} + y^{3} = 12 x y - 8, (6, 4)

\frac{( s + 2 )}{( s ^{2} + 2 s + 2 )}

\int_{0}^{1} x \frac{2 ( x + 2 )}{5} d x

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

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