`int dx / sqrt(1-(x+1)^2)` Find the indefinite integral

Indefinite integral are written in the form of `int f(x) dx = F(x) +C`

 where: f(x) as the integrand

           F(x) as the anti-derivative function 

           C  as the arbitrary constant known as constant of integration

For the given problem, the integrand `f(x) =1/sqrt(1 -(x+1)^2)`  we apply

u-substitution by letting `u =(x+1) `  and `du = 1 dx or du= dx` .

`int (dx)/sqrt(1 -(x+1)^2) = int (du)/sqrt(1 -u^2)`

`int (du)/sqrt(1 -u^2) `  resembles the basic integration` ` formula for inverse sine function: `int (dx)/sqrt(1-x^2)=arcsin(x) +C` .

By applying the formula, we get:

`int (du)/sqrt(1 -u^2) =arcsin(u) +C`

Then to express it in terms of x, we substitute `u=(x+1)` :

`arcsin(u) +C =arcsin(x+1) +C`