What is the integral of (5x+2)/((x+1)(x+4)) ?

The integral of (5x+2)/((x+1)(x+4)) is -ln|x+1|+6ln|x+4|+C

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integral of (5x+2)/((x+1)(x+4))

\int \frac{5 x + 2}{( x + 1 ) ( x + 4 )} d x

- ln ∣ x + 1 ∣ + 6 ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{5 x + 2}{( x + 1 ) ( x + 4 )} d x

Take the partial fraction of

\frac{5 x + 2}{( x + 1 ) ( x + 4 )} : - \frac{1}{x + 1} + \frac{6}{x + 4}

= \int - \frac{1}{x + 1} + \frac{6}{x + 4} 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{1}{x + 1} d x + \int \frac{6}{x + 4} d x

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

\int \frac{6}{x + 4} d x = 6 ln ∣ x + 4 ∣

= - ln ∣ x + 1 ∣ + 6 ln ∣ x + 4 ∣

Add a constant to the solution

= - ln ∣ x + 1 ∣ + 6 ln ∣ x + 4 ∣ + C

What is the integral of (5x+2)/((x+1)(x+4)) ?
The integral of (5x+2)/((x+1)(x+4)) is -ln|x+1|+6ln|x+4|+C