How can I integrate sin(x)/(1-x) using the substitution method?

[NINHARDCOREFAN]

What technique should I use to integrate this: sin(x)/(1-x)?

 

[amcavoy]

What technique should I use to integrate this: sin(x)/(1-x)?

Same one you'd use to integrate sin(x)/x. It can't be done analytically.

 

[quicknote]

One possible technique to integrate this expression is using the substitution method. Let u = 1-x, then du = -dx. Substituting these into the integral yields:

∫ sin(x)/(1-x) dx = -∫ sin(x)/u du

= -∫ sin(x) du/u

= -ln(u) + C

= -ln(1-x) + C

= ln(1-x)^-1 + C

= ln(1/(1-x)) + C

= ln(1+x) + C

Therefore, the integral of sin(x)/(1-x) is ln(1+x) + C.