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

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

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

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

4 (1 + x - ln 1 + x) - x + C

Solution steps

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

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

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

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

\int 1 d x = x

= 4 (1 + x - ln 1 + x) - x

Add a constant to the solution

= 4 (1 + x - ln 1 + x) - x + C

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