Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 36

Recall that indefinite integral follows: int f(x) dx = F(x)+C
where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as constant of integration.
To evaluate given integral problem: int cos(x)/sqrt(sin^2(x)+1)dx or int (cos(x)dx)/sqrt(sin^2(x)+1) , we may apply u-substitution by letting:
u = sin(x) then du =cos(x) dx .
Plug-in the values , the integral becomes:
int (cos(x)dx)/sqrt(sin^2(x)+1)=int (du)/sqrt(u^2+1) or int (du)/sqrt(u^2+1^2)
The integral resembles one of the formulas from the integration table for rational function with roots. We follow:
int (dx)/sqrt(x^2+a^2) = ln|x+sqrt(x^2+a^2)|+C
By comparing x^2+a^2 with u^2+1^2 , we determine the corresponding values as: x=u and a=1.
Applying the values on the integral formula for rational function with roots, we get:
int (du)/sqrt(u^2+1^2)=ln|u+sqrt(u^2+1^2)| +C
=ln|u+sqrt(u^2+1)| +C
Plug-in u = sin(x) on ln|u+sqrt(u^2+1)| +C , we get the indefinite integral as:
int cos(x)/sqrt(sin^2(x)+1)dx=ln|sin(x)+sqrt(sin^2(x)+1)| +C

Aside from this, we can also consider the another formula from integration table:
int 1/sqrt(u^2+1)du = arcsinh(u) +C
Plug-in u = sin(x) on arcsinh(u) +C , we get another form of indefinite integral as:
int cos(x)/sqrt(sin^2(x)+1)dx=arcsinh(sin(x)) +C