What is the integral of (2-sqrt(z))/(2+sqrt(z)) ?

The integral of (2-sqrt(z))/(2+sqrt(z)) is -z+8sqrt(z)-16ln|sqrt(z)+2|+20+C

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integral of (2-sqrt(z))/(2+sqrt(z))

\int \frac{2 - z}{2 + z} d z

- z + 8 z - 16 ln z + 2 + 20 + C

Solution steps

\int \frac{2 - z}{2 + z} d z

Expand

\frac{2 - z}{2 + z} : \frac{2}{2 + z} - \frac{z}{2 + z}

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{2}{2 + z} d z - \int \frac{z}{2 + z} d z

\int \frac{2}{2 + z} d z = 4 (2 + z - 2 ln 2 + z)

\int \frac{z}{2 + z} d z = z - 4 z + 8 ln z + 2 - 12

= 4 (2 + z - 2 ln 2 + z) - (z - 4 z + 8 ln z + 2 - 12)

Simplify

4 (2 + z - 2 ln 2 + z) - (z - 4 z + 8 ln z + 2 - 12) : - z + 8 z - 16 ln z + 2 + 20

= - z + 8 z - 16 ln z + 2 + 20

Add a constant to the solution

= - z + 8 z - 16 ln z + 2 + 20 + C

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What is the integral of (2-sqrt(z))/(2+sqrt(z)) ?
The integral of (2-sqrt(z))/(2+sqrt(z)) is -z+8sqrt(z)-16ln|sqrt(z)+2|+20+C