What is the integral of (2x^2+x+9)/((x+1)(x^2+4)) ?

The integral of (2x^2+x+9)/((x+1)(x^2+4)) is 2ln|x+1|+1/2 arctan(x/2)+C

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integral of (2x^2+x+9)/((x+1)(x^2+4))

\int \frac{2 x ^{2} + x + 9}{( x + 1 ) ( x ^{2} + 4 )} d x

2 ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{2 x ^{2} + x + 9}{( x + 1 ) ( x ^{2} + 4 )} d x

Take the partial fraction of

\frac{2 x ^{2} + x + 9}{( x + 1 ) ( x ^{2} + 4 )} : \frac{2}{x + 1} + \frac{1}{x ^{2} + 4}

= \int \frac{2}{x + 1} + \frac{1}{x ^{2} + 4} 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{2}{x + 1} d x + \int \frac{1}{x ^{2} + 4} d x

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

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

= 2 ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2})

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2}) + C

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What is the integral of (2x^2+x+9)/((x+1)(x^2+4)) ?
The integral of (2x^2+x+9)/((x+1)(x^2+4)) is 2ln|x+1|+1/2 arctan(x/2)+C