Integrate (((cos^2(x)*sin(x)/(1-sin(x)))-sin(x))dx

Question

thanks

Answer 1

If (((cos^2(x)*sin(x)/(1-sin(x)))-sin(x)) mean:

cos ^ 2 (x) * sin (x) / [ 1 - sin(x) ] - sin (x)

then:

cos ^ 2 (x) * sin (x) / [ 1 - sin(x) ] - sin (x) =

sin (x) * { cos ^ 2 (x) / [ 1 - sin(x) ] - 1 } =

sin (x) * { [ 1 - sin ^ 2 (x) ] / [ 1 - sin(x) ] - 1 } =

sin (x) * { [ 1 + sin (x) ] * [ 1 - sin(x) ] / [ 1 - sin(x) ] - 1 } =

sin (x) * [ 1 + sin(x) - 1 ] =

sin (x) * sin (x) = sin ^ 2 (x)


cos ^ 2 (x) * sin (x) / ( 1 - sin(x) ) - sin (x) = sin ^ 2 (x)

Integrate [ cos ^ 2 (x) * sin (x) / ( 1 - sin(x) ) - sin (x) ] dx = Integrate sin ^ 2 (x) dx = x / 2 + sin (2x) / 4 + C

Answer 2

Hmmm. I get x/2 - sin(2x)/4

Answer 3

My typo.

Integrate sin ^ 2 (x) dx = x / 2 - sin (2x) / 4 + C