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

The integral of (sqrt(x)-1)/(sqrt(x)+1) is x-4sqrt(x)+4ln|sqrt(x)+1|-6+C

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

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

x - 4 x + 4 ln x + 1 - 6 + C

Solution steps

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

\frac{x - 1}{x + 1} = \frac{2 x}{x + 1} - 1

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

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

\int \frac{2 x}{x + 1} d x = 4 (\frac{1}{2} (x + 1)^{2} - 2 (x + 1) + ln x + 1)

\int 1 d x = x

= 4 (\frac{1}{2} (x + 1)^{2} - 2 (x + 1) + ln x + 1) - x

Simplify

4 (\frac{1}{2} (x + 1)^{2} - 2 (x + 1) + ln x + 1) - x : x - 4 x + 4 ln x + 1 - 6

= x - 4 x + 4 ln x + 1 - 6

Add a constant to the solution

= x - 4 x + 4 ln x + 1 - 6 + C

What is the integral of (sqrt(x)-1)/(sqrt(x)+1) ?
The integral of (sqrt(x)-1)/(sqrt(x)+1) is x-4sqrt(x)+4ln|sqrt(x)+1|-6+C