What is the integral of (5x^2+3x+25)/((x^2+9)(2x-1)) ?

The integral of (5x^2+3x+25)/((x^2+9)(2x-1)) is 1/2 ln|(x^2)/9+1|+2/3 arctan(x/3)+3/2 ln|2x-1|+C

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integral of (5x^2+3x+25)/((x^2+9)(2x-1))

\int \frac{5 x ^{2} + 3 x + 25}{( x ^{2} + 9 ) ( 2 x - 1 )} d x

\frac{1}{2} ln \frac{x ^{2}}{9} + 1 + \frac{2}{3} arctan (\frac{x}{3}) + \frac{3}{2} ln ∣ 2 x - 1 ∣ + C

Solution steps

\int \frac{5 x ^{2} + 3 x + 25}{( x ^{2} + 9 ) ( 2 x - 1 )} d x

Take the partial fraction of

\frac{5 x ^{2} + 3 x + 25}{( x ^{2} + 9 ) ( 2 x - 1 )} : \frac{x + 2}{x ^{2} + 9} + \frac{3}{2 x - 1}

= \int \frac{x + 2}{x ^{2} + 9} + \frac{3}{2 x - 1} 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 \frac{x + 2}{x ^{2} + 9} d x + \int \frac{3}{2 x - 1} d x

\int \frac{x + 2}{x ^{2} + 9} d x = \frac{1}{2} ln \frac{x ^{2}}{9} + 1 + \frac{2}{3} arctan (\frac{x}{3})

\int \frac{3}{2 x - 1} d x = \frac{3}{2} ln ∣ 2 x - 1 ∣

= \frac{1}{2} ln \frac{x ^{2}}{9} + 1 + \frac{2}{3} arctan (\frac{x}{3}) + \frac{3}{2} ln ∣ 2 x - 1 ∣

Add a constant to the solution

= \frac{1}{2} ln \frac{x ^{2}}{9} + 1 + \frac{2}{3} arctan (\frac{x}{3}) + \frac{3}{2} ln ∣ 2 x - 1 ∣ + C

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What is the integral of (5x^2+3x+25)/((x^2+9)(2x-1)) ?
The integral of (5x^2+3x+25)/((x^2+9)(2x-1)) is 1/2 ln|(x^2)/9+1|+2/3 arctan(x/3)+3/2 ln|2x-1|+C