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

The integral of (x+9)/(2x^2+x-1) is 19/6 ln|2x-1|-8/3 ln|x+1|+C

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

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

\frac{19}{6} ln ∣ 2 x - 1 ∣ - \frac{8}{3} ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x + 9}{2 x ^{2} + x - 1} : \frac{19}{3 ( 2 x - 1 )} - \frac{8}{3 ( x + 1 )}

= \int \frac{19}{3 ( 2 x - 1 )} - \frac{8}{3 ( 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{19}{3 ( 2 x - 1 )} d x - \int \frac{8}{3 ( x + 1 )} d x

\int \frac{19}{3 ( 2 x - 1 )} d x = \frac{19}{6} ln ∣ 2 x - 1 ∣

\int \frac{8}{3 ( x + 1 )} d x = \frac{8}{3} ln ∣ x + 1 ∣

= \frac{19}{6} ln ∣ 2 x - 1 ∣ - \frac{8}{3} ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{19}{6} ln ∣ 2 x - 1 ∣ - \frac{8}{3} ln ∣ x + 1 ∣ + C

What is the integral of (x+9)/(2x^2+x-1) ?
The integral of (x+9)/(2x^2+x-1) is 19/6 ln|2x-1|-8/3 ln|x+1|+C