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

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

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

\int \frac{x + 4}{5 x + 5} d x

\frac{1}{5} (x + 1 - ln (5 ∣ x + 1 ∣)) + \frac{4}{5} ln ∣ 5 x + 5 ∣ + C

Solution steps

\int \frac{x + 4}{5 x + 5} d x

Expand

\frac{x + 4}{5 x + 5} : \frac{x}{5 x + 5} + \frac{4}{5 x + 5}

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{x}{5 x + 5} d x + \int \frac{4}{5 x + 5} d x

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

\int \frac{4}{5 x + 5} d x = \frac{4}{5} ln ∣ 5 x + 5 ∣

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

Add a constant to the solution

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

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