What is the indefinite integral of ( sinx + cos x )/ ( sinx -cosx ) ?

To determine the indefinite integral when the integrand is
the given function, we'll use substitution technique.

We'll
change the variable x, substituting the denominator by another variable,
t.

We'll note the denominator sin x - cos x =
t(x)

We'll differentiate the
denominator:

(sin x - cos x)' = [cos x - (-sin
x)]dx

(cos x + sin x)dx
= dt

We'll notice that the numerator of the function is the
result of differentiating the function.

We'll calculate the
integral:

Int f(x) = Int
dt/t

Int dt/t = ln |t| + C

But
t = sin x - cos x

The indefinite integral of
(sinx + cos x)/ (sinx -cosx) is:

Int f(x) = ln|sin x - cos x|
+ C