What is the integral of (6x^2+5x+59)/((x+1)(x^2+9)) ?

The integral of (6x^2+5x+59)/((x+1)(x^2+9)) is 6ln|x+1|+5/3 arctan(x/3)+C

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integral of (6x^2+5x+59)/((x+1)(x^2+9))

\int \frac{6 x ^{2} + 5 x + 59}{( x + 1 ) ( x ^{2} + 9 )} d x

6 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{6 x ^{2} + 5 x + 59}{( x + 1 ) ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{6 x ^{2} + 5 x + 59}{( x + 1 ) ( x ^{2} + 9 )} : \frac{6}{x + 1} + \frac{5}{x ^{2} + 9}

= \int \frac{6}{x + 1} + \frac{5}{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{6}{x + 1} d x + \int \frac{5}{x ^{2} + 9} d x

\int \frac{6}{x + 1} d x = 6 ln ∣ x + 1 ∣

\int \frac{5}{x ^{2} + 9} d x = \frac{5}{3} arctan (\frac{x}{3})

= 6 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3})

Add a constant to the solution

= 6 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3}) + C

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What is the integral of (6x^2+5x+59)/((x+1)(x^2+9)) ?
The integral of (6x^2+5x+59)/((x+1)(x^2+9)) is 6ln|x+1|+5/3 arctan(x/3)+C