What is the integral of (4x^3-7x-12)/(x^3-x^2-6x) ?

The integral of (4x^3-7x-12)/(x^3-x^2-6x) is

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integral of (4x^3-7x-12)/(x^3-x^2-6x)

\int \frac{4 x ^{3} - 7 x - 12}{x ^{3} - x ^{2} - 6 x} d x

4 x + \frac{1}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x - 3 ∣ - \frac{16}{5} ln ∣ 2 x + 4 ∣ + \frac{36}{5} ln ∣ 2 x - 6 ∣ + 2 ln ∣ x ∣ - 2 + C

Solution steps

\int \frac{4 x ^{3} - 7 x - 12}{x ^{3} - x ^{2} - 6 x} d x

Expand

\frac{4 x ^{3} - 7 x - 12}{x ^{3} - x ^{2} - 6 x} : \frac{4 x ^{3}}{x ^{3} - x ^{2} - 6 x} - \frac{7 x}{x ^{3} - x ^{2} - 6 x} - \frac{12}{x ^{3} - x ^{2} - 6 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{4 x ^{3}}{x ^{3} - x ^{2} - 6 x} d x - \int \frac{7 x}{x ^{3} - x ^{2} - 6 x} d x - \int \frac{12}{x ^{3} - x ^{2} - 6 x} d x

\int \frac{4 x ^{3}}{x ^{3} - x ^{2} - 6 x} d x = 4 (x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣)

\int \frac{7 x}{x ^{3} - x ^{2} - 6 x} d x = 7 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

\int \frac{12}{x ^{3} - x ^{2} - 6 x} d x = 12 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣)

= 4 (x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣) - 7 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 12 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣)

Simplify

4 (x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣) - 7 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 12 (- \frac{1}{6} ln ∣ x ∣ + \frac{1}{10} ln ∣ x + 2 ∣ + \frac{1}{15} ln ∣ x - 3 ∣) : 4 x + \frac{1}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x - 3 ∣ - \frac{16}{5} ln ∣ 2 x + 4 ∣ + \frac{36}{5} ln ∣ 2 x - 6 ∣ + 2 ln ∣ x ∣ - 2

= 4 x + \frac{1}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x - 3 ∣ - \frac{16}{5} ln ∣ 2 x + 4 ∣ + \frac{36}{5} ln ∣ 2 x - 6 ∣ + 2 ln ∣ x ∣ - 2

Add a constant to the solution

= 4 x + \frac{1}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x - 3 ∣ - \frac{16}{5} ln ∣ 2 x + 4 ∣ + \frac{36}{5} ln ∣ 2 x - 6 ∣ + 2 ln ∣ x ∣ - 2 + C

\int 15 sin^{2} (x) cos^{3} (x) d x

\frac{d}{d x} (\frac{7}{2 x})

x \to - 1 lim (4 x^{3} + x^{2} + 2 x + 1)

\frac{d y}{d x} = y x^{9} sin (x^{10})

\frac{d}{d x} (\frac{1}{7 x})

What is the integral of (4x^3-7x-12)/(x^3-x^2-6x) ?
The integral of (4x^3-7x-12)/(x^3-x^2-6x) is