What is the integral of (4x^2+21x+54)/(x^2+6x+13) ?

The integral of (4x^2+21x+54)/(x^2+6x+13) is 4x-3/2 ln|x^2+6x+13|+11/2 arctan(1/2 (x+3))+12+C

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integral of (4x^2+21x+54)/(x^2+6x+13)

\int \frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} d x

4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12 + C

Solution steps

\int \frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} d x

Expand

\frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} : \frac{4 x ^{2}}{x ^{2} + 6 x + 13} + \frac{21 x}{x ^{2} + 6 x + 13} + \frac{54}{x ^{2} + 6 x + 13}

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 ^{2}}{x ^{2} + 6 x + 13} d x + \int \frac{21 x}{x ^{2} + 6 x + 13} d x + \int \frac{54}{x ^{2} + 6 x + 13} d x

\int \frac{4 x ^{2}}{x ^{2} + 6 x + 13} d x = 4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3)))

\int \frac{21 x}{x ^{2} + 6 x + 13} d x = 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)))

\int \frac{54}{x ^{2} + 6 x + 13} d x = 27 arctan (\frac{1}{2} (x + 3))

= 4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3))) + 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))) + 27 arctan (\frac{1}{2} (x + 3))

Simplify

4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3))) + 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))) + 27 arctan (\frac{1}{2} (x + 3)) : 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12

= 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12

Add a constant to the solution

= 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12 + C

x \to 8 lim (6 x^{2} - 9 x + 1)

(\frac{3}{2})^{x}

y^{^{''}} - 4 y^{^{'}} + 2 y = 0, y (0) = 0, y^{^{'}} (0) = 1

\frac{\partial}{\partial y} (4 e^{x} ln (y))

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

What is the integral of (4x^2+21x+54)/(x^2+6x+13) ?
The integral of (4x^2+21x+54)/(x^2+6x+13) is 4x-3/2 ln|x^2+6x+13|+11/2 arctan(1/2 (x+3))+12+C