What is the integral of (5x+4)/((x-3)(x-7)) ?

The integral of (5x+4)/((x-3)(x-7)) is -19/4 ln|x-3|+39/4 ln|x-7|+C

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integral of (5x+4)/((x-3)(x-7))

\int \frac{5 x + 4}{( x - 3 ) ( x - 7 )} d x

- \frac{19}{4} ln ∣ x - 3 ∣ + \frac{39}{4} ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{5 x + 4}{( x - 3 ) ( x - 7 )} d x

Expand

\frac{5 x + 4}{( x - 3 ) ( x - 7 )} : \frac{5 x}{( x - 3 ) ( x - 7 )} + \frac{4}{( x - 3 ) ( x - 7 )}

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{5 x}{( x - 3 ) ( x - 7 )} d x + \int \frac{4}{( x - 3 ) ( x - 7 )} d x

\int \frac{5 x}{( x - 3 ) ( x - 7 )} d x = 5 (ln ∣ x - 7 ∣ + 3 (- \frac{1}{4} ln ∣ x - 3 ∣ + \frac{1}{4} ln ∣ x - 7 ∣))

\int \frac{4}{( x - 3 ) ( x - 7 )} d x = - ln ∣ x - 3 ∣ + ln ∣ x - 7 ∣

= 5 (ln ∣ x - 7 ∣ + 3 (- \frac{1}{4} ln ∣ x - 3 ∣ + \frac{1}{4} ln ∣ x - 7 ∣)) - ln ∣ x - 3 ∣ + ln ∣ x - 7 ∣

Simplify

5 (ln ∣ x - 7 ∣ + 3 (- \frac{1}{4} ln ∣ x - 3 ∣ + \frac{1}{4} ln ∣ x - 7 ∣)) - ln ∣ x - 3 ∣ + ln ∣ x - 7 ∣ : - \frac{19}{4} ln ∣ x - 3 ∣ + \frac{39}{4} ln ∣ x - 7 ∣

= - \frac{19}{4} ln ∣ x - 3 ∣ + \frac{39}{4} ln ∣ x - 7 ∣

Add a constant to the solution

= - \frac{19}{4} ln ∣ x - 3 ∣ + \frac{39}{4} ln ∣ x - 7 ∣ + C

What is the integral of (5x+4)/((x-3)(x-7)) ?
The integral of (5x+4)/((x-3)(x-7)) is -19/4 ln|x-3|+39/4 ln|x-7|+C