What is the integral of (x^3+7x^2+25x+35)/(x^2+5x+6) ?

The integral of (x^3+7x^2+25x+35)/(x^2+5x+6) is

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integral of (x^3+7x^2+25x+35)/(x^2+5x+6)

\int \frac{x ^{3} + 7 x ^{2} + 25 x + 35}{x ^{2} + 5 x + 6} d x

\frac{1}{2} x^{2} + 2 x + \frac{19}{2} ln (4 x^{2} + 5 x + 6) + \frac{35}{2} ln (2 ∣ x + 3 ∣) - \frac{35}{2} ln (2 ∣ x + 2 ∣) - 5 ln 4 x^{2} + 20 x + 24 + 17 ln ∣ 2 x + 6 ∣ - 17 ln ∣ 2 x + 4 ∣ + 35 ln ∣ x + 2 ∣ - 35 ln ∣ x + 3 ∣ + \frac{35}{2} - \frac{125}{8} + C

Solution steps

\int \frac{x ^{3} + 7 x ^{2} + 25 x + 35}{x ^{2} + 5 x + 6} d x

Expand

\frac{x ^{3} + 7 x ^{2} + 25 x + 35}{x ^{2} + 5 x + 6} : \frac{x ^{3}}{x ^{2} + 5 x + 6} + \frac{7 x ^{2}}{x ^{2} + 5 x + 6} + \frac{25 x}{x ^{2} + 5 x + 6} + \frac{35}{x ^{2} + 5 x + 6}

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{x ^{3}}{x ^{2} + 5 x + 6} d x + \int \frac{7 x ^{2}}{x ^{2} + 5 x + 6} d x + \int \frac{25 x}{x ^{2} + 5 x + 6} d x + \int \frac{35}{x ^{2} + 5 x + 6} d x

\int \frac{x ^{3}}{x ^{2} + 5 x + 6} d x = \frac{1}{8} (4 x^{2} - 40 x + 76 ln (4 x^{2} + 5 x + 6) + 140 ln (2 ∣ x + 3 ∣) - 140 ln (2 ∣ x + 2 ∣) - 125)

\int \frac{7 x ^{2}}{x ^{2} + 5 x + 6} d x = 7 (x + \frac{5}{2} + 4 (- \frac{5}{8} ln 4 x^{2} + 20 x + 24 - \frac{13}{8} ln ∣ 2 x + 6 ∣ + \frac{13}{8} ln ∣ 2 x + 4 ∣))

\int \frac{25 x}{x ^{2} + 5 x + 6} d x = 50 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣))

\int \frac{35}{x ^{2} + 5 x + 6} d x = 35 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣)

= \frac{1}{8} (4 x^{2} - 40 x + 76 ln (4 x^{2} + 5 x + 6) + 140 ln (2 ∣ x + 3 ∣) - 140 ln (2 ∣ x + 2 ∣) - 125) + 7 (x + \frac{5}{2} + 4 (- \frac{5}{8} ln 4 x^{2} + 20 x + 24 - \frac{13}{8} ln ∣ 2 x + 6 ∣ + \frac{13}{8} ln ∣ 2 x + 4 ∣)) + 50 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣)) + 35 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣)

Simplify

\frac{1}{8} (4 x^{2} - 40 x + 76 ln (4 x^{2} + 5 x + 6) + 140 ln (2 ∣ x + 3 ∣) - 140 ln (2 ∣ x + 2 ∣) - 125) + 7 (x + \frac{5}{2} + 4 (- \frac{5}{8} ln 4 x^{2} + 20 x + 24 - \frac{13}{8} ln ∣ 2 x + 6 ∣ + \frac{13}{8} ln ∣ 2 x + 4 ∣)) + 50 (\frac{1}{4} ln 4 x^{2} + 20 x + 24 + \frac{5}{4} (ln ∣ 2 x + 6 ∣ - ln ∣ 2 x + 4 ∣)) + 35 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) : \frac{1}{2} x^{2} + 2 x + \frac{19}{2} ln (4 x^{2} + 5 x + 6) + \frac{35}{2} ln (2 ∣ x + 3 ∣) - \frac{35}{2} ln (2 ∣ x + 2 ∣) - 5 ln 4 x^{2} + 20 x + 24 + 17 ln ∣ 2 x + 6 ∣ - 17 ln ∣ 2 x + 4 ∣ + 35 ln ∣ x + 2 ∣ - 35 ln ∣ x + 3 ∣ + \frac{35}{2} - \frac{125}{8}

= \frac{1}{2} x^{2} + 2 x + \frac{19}{2} ln (4 x^{2} + 5 x + 6) + \frac{35}{2} ln (2 ∣ x + 3 ∣) - \frac{35}{2} ln (2 ∣ x + 2 ∣) - 5 ln 4 x^{2} + 20 x + 24 + 17 ln ∣ 2 x + 6 ∣ - 17 ln ∣ 2 x + 4 ∣ + 35 ln ∣ x + 2 ∣ - 35 ln ∣ x + 3 ∣ + \frac{35}{2} - \frac{125}{8}

Add a constant to the solution

= \frac{1}{2} x^{2} + 2 x + \frac{19}{2} ln (4 x^{2} + 5 x + 6) + \frac{35}{2} ln (2 ∣ x + 3 ∣) - \frac{35}{2} ln (2 ∣ x + 2 ∣) - 5 ln 4 x^{2} + 20 x + 24 + 17 ln ∣ 2 x + 6 ∣ - 17 ln ∣ 2 x + 4 ∣ + 35 ln ∣ x + 2 ∣ - 35 ln ∣ x + 3 ∣ + \frac{35}{2} - \frac{125}{8} + C

\int \frac{4 sinh ( x )}{4 + cosh ^{2} ( x )} d x

\frac{d y}{d x}, y = 3 x

\frac{d}{d x} (x + \frac{9}{x})

\int (3 - 8 x) d x

What is the integral of (x^3+7x^2+25x+35)/(x^2+5x+6) ?
The integral of (x^3+7x^2+25x+35)/(x^2+5x+6) is