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

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

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

\int \frac{x ^{2} - 7 x - 12}{x ( x ^{2} + 3 x + 3 )} d x

- 4 ln ∣ x ∣ + 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{10}{3} arctan (\frac{2}{3} x + 3) + C

Solution steps

\int \frac{x ^{2} - 7 x - 12}{x ( x ^{2} + 3 x + 3 )} d x

Expand

x (x^{2} + 3 x + 3) : x^{3} + 3 x^{2} + 3 x

= \int \frac{x ^{2} - 7 x - 12}{x ^{3} + 3 x ^{2} + 3 x} d x

Take the partial fraction of

\frac{x ^{2} - 7 x - 12}{x ^{3} + 3 x ^{2} + 3 x} : - \frac{4}{x} + \frac{5 x + 5}{x ^{2} + 3 x + 3}

= \int - \frac{4}{x} + \frac{5 x + 5}{x ^{2} + 3 x + 3} 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{4}{x} d x + \int \frac{5 x + 5}{x ^{2} + 3 x + 3} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{5 x + 5}{x ^{2} + 3 x + 3} d x = 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{10}{3} arctan (\frac{2}{3} x + 3)

= - 4 ln ∣ x ∣ + 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{10}{3} arctan (\frac{2}{3} x + 3)

Add a constant to the solution

= - 4 ln ∣ x ∣ + 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{10}{3} arctan (\frac{2}{3} x + 3) + C

\frac{d}{d x} (3 x^{4} - 8 x^{3} + 2)

f (x) = - 5 x^{2} - 6 x + 200

lo g_{6} (x)

y^{^{''}} + 2 t = 0

\int_{- 1}^{2} - x^{2} + x + 2 d x

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