Evaluate∫x−sinx1−cosxdx.
Given to evaluate ∫x−sinx1−cosxdx.
∫x−sinx1−1−2sin2x2dx=∫x−sinx2sin2x2dx=12∫xsin2x2−12∫2sinx2cosx2sin2x2dx=12∫x⋅cosec2x2−∫cotx2dx=12x⋅∫cosec2x2−∫ddxx⋅∫cosec2x2dxdx−ln2sinx2=12−x⋅2cotx2+∫1⋅cotx2⋅2dx−ln2sinx2=12−2xcotx2+2lnsinx2⋅2=12−2xcotx2+ln2⋅sinx2+C
Hence, ∫x−sinx1−cosxdx=12−2xcotx2+ln2⋅sinx2+C.
Evaluate the following expression forx=-1,y=-2,z=3
xy+yz+zx