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

The integral of (p(x))/((x-2)(x^2+9)) is p(2/13 ln|x-2|+1/13 (-ln|x^2+9|+3arctan(x/3)))+C

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

\int \frac{p ( x )}{( x - 2 ) ( x ^{2} + 9 )} d x

p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + C

Solution steps

\int \frac{p x}{( x - 2 ) ( x ^{2} + 9 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= p \cdot \int \frac{x}{( - 2 + x ) ( 9 + x ^{2} )} d x

Take the partial fraction of

\frac{x}{( - 2 + x ) ( 9 + x ^{2} )} : \frac{2}{13 ( x - 2 )} + \frac{- 2 x + 9}{13 ( x ^{2} + 9 )}

= p \cdot \int \frac{2}{13 ( x - 2 )} + \frac{- 2 x + 9}{13 ( 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

= p (\int \frac{2}{13 ( x - 2 )} d x + \int \frac{- 2 x + 9}{13 ( x ^{2} + 9 )} d x)

\int \frac{2}{13 ( x - 2 )} d x = \frac{2}{13} ln ∣ x - 2 ∣

\int \frac{- 2 x + 9}{13 ( x ^{2} + 9 )} d x = \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))

= p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3})))

Add a constant to the solution

= p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + C

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What is the integral of (p(x))/((x-2)(x^2+9)) ?
The integral of (p(x))/((x-2)(x^2+9)) is p(2/13 ln|x-2|+1/13 (-ln|x^2+9|+3arctan(x/3)))+C