What is the integral of (4x^2-14x+6)/((x-9)(x^2-25)) ?

The integral of (4x^2-14x+6)/((x-9)(x^2-25)) is 51/14 ln|x-9|+44/35 ln|x+5|-9/10 ln|x-5|+C

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integral of (4x^2-14x+6)/((x-9)(x^2-25))

\int \frac{4 x ^{2} - 14 x + 6}{( x - 9 ) ( x ^{2} - 25 )} d x

\frac{51}{14} ln ∣ x - 9 ∣ + \frac{44}{35} ln ∣ x + 5 ∣ - \frac{9}{10} ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{4 x ^{2} - 14 x + 6}{( x - 9 ) ( x ^{2} - 25 )} d x

Take the partial fraction of

\frac{4 x ^{2} - 14 x + 6}{( x - 9 ) ( x ^{2} - 25 )} : \frac{51}{14 ( x - 9 )} + \frac{44}{35 ( x + 5 )} - \frac{9}{10 ( x - 5 )}

= \int \frac{51}{14 ( x - 9 )} + \frac{44}{35 ( x + 5 )} - \frac{9}{10 ( x - 5 )} 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{51}{14 ( x - 9 )} d x + \int \frac{44}{35 ( x + 5 )} d x - \int \frac{9}{10 ( x - 5 )} d x

\int \frac{51}{14 ( x - 9 )} d x = \frac{51}{14} ln ∣ x - 9 ∣

\int \frac{44}{35 ( x + 5 )} d x = \frac{44}{35} ln ∣ x + 5 ∣

\int \frac{9}{10 ( x - 5 )} d x = \frac{9}{10} ln ∣ x - 5 ∣

= \frac{51}{14} ln ∣ x - 9 ∣ + \frac{44}{35} ln ∣ x + 5 ∣ - \frac{9}{10} ln ∣ x - 5 ∣

Add a constant to the solution

= \frac{51}{14} ln ∣ x - 9 ∣ + \frac{44}{35} ln ∣ x + 5 ∣ - \frac{9}{10} ln ∣ x - 5 ∣ + C

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What is the integral of (4x^2-14x+6)/((x-9)(x^2-25)) ?
The integral of (4x^2-14x+6)/((x-9)(x^2-25)) is 51/14 ln|x-9|+44/35 ln|x+5|-9/10 ln|x-5|+C