What is the integral of ((2x+5))/(x^2-4x+7) ?

The integral of ((2x+5))/(x^2-4x+7) is ln| 1/3 (x-2)^2+1|+3sqrt(3)arctan(1/(sqrt(3))(x-2))+C

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integral of ((2x+5))/(x^2-4x+7)

\int \frac{( 2 x + 5 )}{x ^{2} - 4 x + 7} d x

ln \frac{1}{3} (x - 2)^{2} + 1 + 33 arctan (\frac{1}{3} (x - 2)) + C

Solution steps

\int \frac{2 x + 5}{x ^{2} - 4 x + 7} d x

Expand

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

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} - 4 x + 7} d x + \int \frac{5}{x ^{2} - 4 x + 7} d x

\int \frac{2 x}{x ^{2} - 4 x + 7} d x = ln \frac{1}{3} (x - 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x - 2))

\int \frac{5}{x ^{2} - 4 x + 7} d x = \frac{5}{3} arctan (\frac{1}{3} (x - 2))

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

Add similar elements:

\frac{4}{3} arctan (\frac{1}{3} (x - 2)) + \frac{5}{3} arctan (\frac{1}{3} (x - 2)) = 33 arctan (\frac{1}{3} (x - 2))

= ln \frac{1}{3} (x - 2)^{2} + 1 + 33 arctan (\frac{1}{3} (x - 2))

Add a constant to the solution

= ln \frac{1}{3} (x - 2)^{2} + 1 + 33 arctan (\frac{1}{3} (x - 2)) + C

What is the integral of ((2x+5))/(x^2-4x+7) ?
The integral of ((2x+5))/(x^2-4x+7) is ln| 1/3 (x-2)^2+1|+3sqrt(3)arctan(1/(sqrt(3))(x-2))+C