What is the integral of (2x^3+6x^2+7x+4)/(x^2+3x+2) ?

The integral of (2x^3+6x^2+7x+4)/(x^2+3x+2) is x^2-3ln|x+1|+6ln|x+2|-4ln|2x+4|+4ln|2x+2|+C

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integral of (2x^3+6x^2+7x+4)/(x^2+3x+2)

\int \frac{2 x ^{3} + 6 x ^{2} + 7 x + 4}{x ^{2} + 3 x + 2} d x

x^{2} - 3 ln ∣ x + 1 ∣ + 6 ln ∣ x + 2 ∣ - 4 ln ∣ 2 x + 4 ∣ + 4 ln ∣ 2 x + 2 ∣ + C

Solution steps

\int \frac{2 x ^{3} + 6 x ^{2} + 7 x + 4}{x ^{2} + 3 x + 2} d x

Expand

\frac{2 x ^{3} + 6 x ^{2} + 7 x + 4}{x ^{2} + 3 x + 2} : \frac{2 x ^{3}}{x ^{2} + 3 x + 2} + \frac{6 x ^{2}}{x ^{2} + 3 x + 2} + \frac{7 x}{x ^{2} + 3 x + 2} + \frac{4}{x ^{2} + 3 x + 2}

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 ^{3}}{x ^{2} + 3 x + 2} d x + \int \frac{6 x ^{2}}{x ^{2} + 3 x + 2} d x + \int \frac{7 x}{x ^{2} + 3 x + 2} d x + \int \frac{4}{x ^{2} + 3 x + 2} d x

\int \frac{2 x ^{3}}{x ^{2} + 3 x + 2} d x = x^{2} - 6 x - 2 ln ∣ x + 1 ∣ + 16 ln ∣ x + 2 ∣

\int \frac{6 x ^{2}}{x ^{2} + 3 x + 2} d x = 6 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x)

\int \frac{7 x}{x ^{2} + 3 x + 2} d x = 7 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣)

\int \frac{4}{x ^{2} + 3 x + 2} d x = - 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣)

= x^{2} - 6 x - 2 ln ∣ x + 1 ∣ + 16 ln ∣ x + 2 ∣ + 6 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x) + 7 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) - 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣)

Simplify

x^{2} - 6 x - 2 ln ∣ x + 1 ∣ + 16 ln ∣ x + 2 ∣ + 6 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x) + 7 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) - 8 (\frac{1}{2} ln ∣ 2 x + 4 ∣ - \frac{1}{2} ln ∣ 2 x + 2 ∣) : x^{2} - 3 ln ∣ x + 1 ∣ + 6 ln ∣ x + 2 ∣ - 4 ln ∣ 2 x + 4 ∣ + 4 ln ∣ 2 x + 2 ∣

= x^{2} - 3 ln ∣ x + 1 ∣ + 6 ln ∣ x + 2 ∣ - 4 ln ∣ 2 x + 4 ∣ + 4 ln ∣ 2 x + 2 ∣

Add a constant to the solution

= x^{2} - 3 ln ∣ x + 1 ∣ + 6 ln ∣ x + 2 ∣ - 4 ln ∣ 2 x + 4 ∣ + 4 ln ∣ 2 x + 2 ∣ + C

y = \frac{1}{x}, (1, 1)

\frac{d}{d x} ((x^{2} + 3 x - 9) e^{x})

\frac{d}{d x} (x^{3} + k y (x)^{2} - 5 x y)

\frac{16 x}{x ^{2} + 16}, at - 2, - \frac{8}{5}

\frac{\partial}{\partial x} (2 e^{x y})

What is the integral of (2x^3+6x^2+7x+4)/(x^2+3x+2) ?
The integral of (2x^3+6x^2+7x+4)/(x^2+3x+2) is x^2-3ln|x+1|+6ln|x+2|-4ln|2x+4|+4ln|2x+2|+C