What is the integral of y/(y+6) ?

The integral of y/(y+6) is y+6-6ln|y+6|+C

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integral of y/(y+6)

\int \frac{y}{y + 6} d y

y + 6 - 6 ln ∣ y + 6 ∣ + C

Solution steps

\int \frac{y}{y + 6} d y

Apply u-substitution

: \int \frac{u - 6}{u} d u

= \int \frac{u - 6}{u} d u

Expand

\frac{u - 6}{u} : 1 - \frac{6}{u}

= \int 1 - \frac{6}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int \frac{6}{u} d u

\int 1 d u = u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= u - 6 ln ∣ u ∣

Substitute back

u = y + 6

= y + 6 - 6 ln ∣ y + 6 ∣

Add a constant to the solution

= y + 6 - 6 ln ∣ y + 6 ∣ + C

\int 2 x (x^{2} - 1)^{2} d x

x = 20 - 5 y^{2}, x = 5 y^{2} - 20

x \to 3 lim (\frac{3}{9 - x ^{2}})

y = x 4 - x^{2}, (2, - 2)