What is the integral of (2x+1)/(x^2-x+3) ?

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

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integral of (2x+1)/(x^2-x+3)

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

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

Solution steps

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

Expand

\frac{2 x + 1}{x ^{2} - x + 3} : \frac{2 x}{x ^{2} - x + 3} + \frac{1}{x ^{2} - x + 3}

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

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

\int \frac{1}{x ^{2} - x + 3} d x = \frac{2}{11} arctan (\frac{2 x - 1}{11})

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

Simplify

ln 4 x^{2} - 4 x + 12 + \frac{2}{11} arctan (\frac{2 x - 1}{11}) + \frac{2}{11} arctan (\frac{2 x - 1}{11}) : ln 4 x^{2} - 4 x + 12 + \frac{4}{11} arctan (\frac{1}{11} (2 x - 1))

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

Add a constant to the solution

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

x \to 2 lim (\frac{8}{x})

\int_{9}^{\infty} \frac{2}{x ^{3}} d x

(y e^{x y} - \frac{1}{y}) d x (x e^{x y} - \frac{x}{y ^{2}}) d y = 0

f (x) = \frac{5 t + 300}{t ^{2} + 25}

2 x^{3} + 4 x

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