What is the integral of (y+4)/(y^2+y) ?

The integral of (y+4)/(y^2+y) is 4ln|y|-3ln|y+1|+C

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integral of (y+4)/(y^2+y)

\int \frac{y + 4}{y ^{2} + y} d y

4 ln ∣ y ∣ - 3 ln ∣ y + 1 ∣ + C

Solution steps

\int \frac{y + 4}{y ^{2} + y} d y

Take the partial fraction of

\frac{y + 4}{y ^{2} + y} : \frac{4}{y} - \frac{3}{y + 1}

= \int \frac{4}{y} - \frac{3}{y + 1} d y

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{4}{y} d y - \int \frac{3}{y + 1} d y

\int \frac{4}{y} d y = 4 ln ∣ y ∣

\int \frac{3}{y + 1} d y = 3 ln ∣ y + 1 ∣

= 4 ln ∣ y ∣ - 3 ln ∣ y + 1 ∣

Add a constant to the solution

= 4 ln ∣ y ∣ - 3 ln ∣ y + 1 ∣ + C

What is the integral of (y+4)/(y^2+y) ?
The integral of (y+4)/(y^2+y) is 4ln|y|-3ln|y+1|+C