What's the indefinite integral of |sin(x)|dx?

Question

In other words, the indefinite integral of ( abs [ sin [ x ] ] dx )? Many thanks!

Answer 1

|-cos(x)| + c

Answer 2

∫ | sinx | dx =

the integrand is actually:

| sinx | = sinx if sinx ≥ 0

| sinx | = - sinx if sinx < 0

hence:

∫ | sinx | dx = ∫ sinx dx = - cosx + C if sinx ≥ 0

∫ | sinx | dx = ∫ - sinx dx = cosx + C if sinx < 0

thus, joining these into a single statement:

∫ | sinx | dx = (- cosx) (| sinx | / sinx) + C

or, more precisely,

∫ | sinx | dx = (- cosx) sign(sinx) + C

sign(sinx) meaning 1 if sinx > 0 and - 1 if sinx < 0

I hope it helps..

Bye!