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

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

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

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

- 2 ln x + 1 + 1 + 2 x + 1 + 2 + C

Solution steps

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

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

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

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

\int 1 d x = x

= - (x + 2 ln x + 1 + 1 - 2 x + 1 - 2) + x

Simplify

= - 2 ln x + 1 + 1 + 2 x + 1 + 2

Add a constant to the solution

= - 2 ln x + 1 + 1 + 2 x + 1 + 2 + C

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