What is the integral of (4x+7)/((3x+2)(x-5)) ?

The integral of (4x+7)/((3x+2)(x-5)) is -13/51 ln|3x+2|+27/17 ln|x-5|+C

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integral of (4x+7)/((3x+2)(x-5))

\int \frac{4 x + 7}{( 3 x + 2 ) ( x - 5 )} d x

- \frac{13}{51} ln ∣ 3 x + 2 ∣ + \frac{27}{17} ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{4 x + 7}{( 3 x + 2 ) ( x - 5 )} d x

Take the partial fraction of

\frac{4 x + 7}{( 3 x + 2 ) ( x - 5 )} : - \frac{13}{17 ( 3 x + 2 )} + \frac{27}{17 ( x - 5 )}

= \int - \frac{13}{17 ( 3 x + 2 )} + \frac{27}{17 ( 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{13}{17 ( 3 x + 2 )} d x + \int \frac{27}{17 ( x - 5 )} d x

\int \frac{13}{17 ( 3 x + 2 )} d x = \frac{13}{51} ln ∣ 3 x + 2 ∣

\int \frac{27}{17 ( x - 5 )} d x = \frac{27}{17} ln ∣ x - 5 ∣

= - \frac{13}{51} ln ∣ 3 x + 2 ∣ + \frac{27}{17} ln ∣ x - 5 ∣

Add a constant to the solution

= - \frac{13}{51} ln ∣ 3 x + 2 ∣ + \frac{27}{17} ln ∣ x - 5 ∣ + C

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What is the integral of (4x+7)/((3x+2)(x-5)) ?
The integral of (4x+7)/((3x+2)(x-5)) is -13/51 ln|3x+2|+27/17 ln|x-5|+C