What is the integral of (3x-11)/((x+4)(5x-1)) ?

The integral of (3x-11)/((x+4)(5x-1)) is 23/21 ln|x+4|-52/105 ln|5x-1|+C

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integral of (3x-11)/((x+4)(5x-1))

\int \frac{3 x - 11}{( x + 4 ) ( 5 x - 1 )} d x

\frac{23}{21} ln ∣ x + 4 ∣ - \frac{52}{105} ln ∣ 5 x - 1 ∣ + C

Solution steps

\int \frac{3 x - 11}{( x + 4 ) ( 5 x - 1 )} d x

Take the partial fraction of

\frac{3 x - 11}{( x + 4 ) ( 5 x - 1 )} : \frac{23}{21 ( x + 4 )} - \frac{52}{21 ( 5 x - 1 )}

= \int \frac{23}{21 ( x + 4 )} - \frac{52}{21 ( 5 x - 1 )} 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{23}{21 ( x + 4 )} d x - \int \frac{52}{21 ( 5 x - 1 )} d x

\int \frac{23}{21 ( x + 4 )} d x = \frac{23}{21} ln ∣ x + 4 ∣

\int \frac{52}{21 ( 5 x - 1 )} d x = \frac{52}{105} ln ∣ 5 x - 1 ∣

= \frac{23}{21} ln ∣ x + 4 ∣ - \frac{52}{105} ln ∣ 5 x - 1 ∣

Add a constant to the solution

= \frac{23}{21} ln ∣ x + 4 ∣ - \frac{52}{105} ln ∣ 5 x - 1 ∣ + C

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What is the integral of (3x-11)/((x+4)(5x-1)) ?
The integral of (3x-11)/((x+4)(5x-1)) is 23/21 ln|x+4|-52/105 ln|5x-1|+C