What is the integral of (2-sqrt(x+1))/(2+sqrt(x+1)) ?

The integral of (2-sqrt(x+1))/(2+sqrt(x+1)) is -x+8sqrt(x+1)-16ln|sqrt(x+1)+2|+19+C

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integral of (2-sqrt(x+1))/(2+sqrt(x+1))

\int \frac{2 - x + 1}{2 + x + 1} d x

- x + 8 x + 1 - 16 ln x + 1 + 2 + 19 + C

Solution steps

\int \frac{2 - x + 1}{2 + x + 1} d x

Apply u-substitution

: \int \frac{2 - u}{2 + u} d u

= \int \frac{2 - u}{2 + u} d u

Expand

\frac{2 - u}{2 + u} : \frac{2}{2 + u} - \frac{u}{2 + 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 \frac{2}{2 + u} d u - \int \frac{u}{2 + u} d u

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

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

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

Substitute back

u = x + 1

= 4 (2 + x + 1 - 2 ln 2 + x + 1) - (x + 1 - 4 x + 1 + 8 ln x + 1 + 2 - 12)

Simplify

4 (2 + x + 1 - 2 ln 2 + x + 1) - (x + 1 - 4 x + 1 + 8 ln x + 1 + 2 - 12) : - x + 8 x + 1 - 16 ln x + 1 + 2 + 19

= - x + 8 x + 1 - 16 ln x + 1 + 2 + 19

Add a constant to the solution

= - x + 8 x + 1 - 16 ln x + 1 + 2 + 19 + C

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3 x^{2}

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What is the integral of (2-sqrt(x+1))/(2+sqrt(x+1)) ?
The integral of (2-sqrt(x+1))/(2+sqrt(x+1)) is -x+8sqrt(x+1)-16ln|sqrt(x+1)+2|+19+C