What is the integral of (2x^2+3x+21)/((x+1)(x^2+9)) ?

The integral of (2x^2+3x+21)/((x+1)(x^2+9)) is 2ln|x+1|+arctan(x/3)+C

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integral of (2x^2+3x+21)/((x+1)(x^2+9))

\int \frac{2 x ^{2} + 3 x + 21}{( x + 1 ) ( x ^{2} + 9 )} d x

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

Solution steps

\int \frac{2 x ^{2} + 3 x + 21}{( x + 1 ) ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{2 x ^{2} + 3 x + 21}{( x + 1 ) ( x ^{2} + 9 )} : \frac{2}{x + 1} + \frac{3}{x ^{2} + 9}

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

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

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

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

Add a constant to the solution

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

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What is the integral of (2x^2+3x+21)/((x+1)(x^2+9)) ?
The integral of (2x^2+3x+21)/((x+1)(x^2+9)) is 2ln|x+1|+arctan(x/3)+C