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

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

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

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

2 x - 19 ln ∣ x + 3 ∣ + 9 ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int 2 - \frac{19}{x + 3} + \frac{9}{x + 2} 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 2 d x - \int \frac{19}{x + 3} d x + \int \frac{9}{x + 2} d x

\int 2 d x = 2 x

\int \frac{19}{x + 3} d x = 19 ln ∣ x + 3 ∣

\int \frac{9}{x + 2} d x = 9 ln ∣ x + 2 ∣

= 2 x - 19 ln ∣ x + 3 ∣ + 9 ln ∣ x + 2 ∣

Add a constant to the solution

= 2 x - 19 ln ∣ x + 3 ∣ + 9 ln ∣ x + 2 ∣ + C

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