Calculus of a Single Variable, Chapter 5, 5.7, Section 5.7, Problem 5

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