Integration :)

Take the integral:
 integral(sin(x))/(sin(x) + cos(x)) dx

Multiply numerator and denominator of (sin(x))/(sin(x) + cos(x)) by csc^3(x):
 = integral(csc^2(x))/(csc^2(x) + cot(x) csc^2(x)) dx

Prepare to substitute u = cot(x). Rewrite (csc^2(x))/(csc^2(x) + cot(x) csc^2(x)) using csc^2(x) = cot^2(x) + 1:
 = integral(csc^2(x))/(cot^3(x) + cot^2(x) + cot(x) + 1) dx
For the integrand (csc^2(x))/(cot^3(x) + cot^2(x) + cot(x) + 1), substitute u = cot(x) and du = -csc^2(x) dx:
 = integral-1/(u^3 + u^2 + u + 1) du

Factor out constants:
 = - integral1/(u^3 + u^2 + u + 1) du

For the integrand 1/(u^3 + u^2 + u + 1), use partial fractions:
 = - integral((1 - u)/(2 (u^2 + 1)) + 1/(2 (u + 1))) du

Integrate the sum term by term and factor out constants:
 = -1/2 integral(1 - u)/(u^2 + 1) du - 1/2 integral1/(u + 1) du

Expanding the integrand (1 - u)/(u^2 + 1) gives 1/(u^2 + 1) - u/(u^2 + 1):
 = -1/2 integral(1/(u^2 + 1) - u/(u^2 + 1)) du - 1/2 integral1/(u + 1) du

Integrate the sum term by term and factor out constants:
 = 1/2 integral u/(u^2 + 1) du - 1/2 integral1/(u^2 + 1) du - 1/2 integral1/(u + 1) du

For the integrand u/(u^2 + 1), substitute s = u^2 + 1 and ds = 2 u du:
 = 1/4 integral1/s ds - 1/2 integral1/(u^2 + 1) du - 1/2 integral1/(u + 1) du
The integral of 1/s is log(s):
 = (log(s))/4 - 1/2 integral1/(u^2 + 1) du - 1/2 integral1/(u + 1) du

The integral of 1/(u^2 + 1) is tan^(-1)(u):
 = -1/2 tan^(-1)(u) + (log(s))/4 - 1/2 integral1/(u + 1) du
For the integrand 1/(u + 1), substitute p = u + 1 and dp = du:
 = -1/2 tan^(-1)(u) + (log(s))/4 - 1/2 integral1/p dp

The integral of 1/p is log(p):
 = -(log(p))/2 + (log(s))/4 - 1/2 tan^(-1)(u) + constant
Substitute back for p = u + 1:

 = (log(s))/4 - 1/2 log(u + 1) - 1/2 tan^(-1)(u) + constant
Substitute back for s = u^2 + 1:
 = 1/4 log(u^2 + 1) - 1/2 log(u + 1) - 1/2 tan^(-1)(u) + constant

Substitute back for u = cot(x):
 = -1/2 log(cot(x) + 1) - 1/2 tan^(-1)(cot(x)) + 1/4 log(csc^2(x)) + constant
Factor the answer a different way:
 = 1/4 (-2 log(cot(x) + 1) - 2 tan^(-1)(cot(x)) + log(csc^2(x))) + constant
Which is equivalent for restricted x values to:
Answer: | = 1/2 (x - log(sin(x) + cos(x))) + constant

 

Sorry Max: I'm too old to learn LaTex! I hope you can follow it.