Let us differentiate all the options one by one to get the expression in the question whose integral is to be found. Here xesin‌x is the common term in all the options. So,let us differentiate it first. Let l=xesin‌x ⇒‌‌
dl
dx
=esinx[x‌cosx+1] ⇒‌‌
dl
dx
=‌‌
esinx
cos2x
[xcos3x+cos2x] Let m=sec‌xesin‌x
⇒‌‌
dm
dx
=secxesinx⋅cosx+esinx‌secx‌tanx
⇒‌‌
dm
dx
=esinx[1+‌‌
sinx
cos2x
] ⇒‌‌
dm
dx
=‌‌
esinx
cos2x
[cos2x+sinx] Differentiation of option (a) is
=‌‌
esinx
cos2x
[xcos3x+cos2x+cos2x+sinx]
=‌‌
esinx
cos2x
[xcos3x+2cos2x+sinx] Differentiation of option (b) is