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

The integral of (x^3+4x^2+7x+6)/(x^2+3x+4) is (x^2)/2+x+4/(sqrt(7))arctan((2x+3)/(sqrt(7)))+C

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

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

\frac{x ^{2}}{2} + x + \frac{4}{7} arctan (\frac{2 x + 3}{7}) + C

Solution steps

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

Long division

\frac{x ^{3} + 4 x ^{2} + 7 x + 6}{x ^{2} + 3 x + 4} : x + 1 + \frac{2}{x ^{2} + 3 x + 4}

= \int x + 1 + \frac{2}{x ^{2} + 3 x + 4} 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 x d x + \int 1 d x + \int \frac{2}{x ^{2} + 3 x + 4} d x

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

\int \frac{2}{x ^{2} + 3 x + 4} d x = \frac{4}{7} arctan (\frac{2 x + 3}{7})

= \frac{x ^{2}}{2} + x + \frac{4}{7} arctan (\frac{2 x + 3}{7})

Add a constant to the solution

= \frac{x ^{2}}{2} + x + \frac{4}{7} arctan (\frac{2 x + 3}{7}) + C

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