How do you integrate the indefinite integral of |sin(x)|dx?

Question

How do you integrate the indefinite integral of the absolute value of sin(x)? I'd like to know exactly how and why. I've come across multiple answers, and I was just wondering which is right.

Method 1: Keep absolute value outside integral, take anti-derivative of sin(x) and apply the absolute value afterwards. Solution: |-cos(x)|+C

Method 2: Add integrals of +sin(x) and -sin(x). Solution: (-cos(x)+C)+(cos(x)+C)

Method 3: Subtract integral of -sin(x) from +sin(x). Solution (-cos(x)+C)-(cos(x)+C)

If there is the RIGHT method, and it's not above, I'd like to know. Also, on method 3, how would that simplify? -2cos(x)?

Answer 1

it's easy!
integrate |sin(x)|dx=-((-1)^int(x/%pi))*cos(x)+2*int(x/%pi)+C
where C is an arbitary constant.