If ∫sec2x−2010sin2010x dx=P(x)(sin x)2010+C
P(π3)=√3
∫sec2x−2010sin2010xdx
=∫sec2xsin2010xdx−∫2010sin2010xdx
=tan x(sin x)2010+2010∫tan x cos x(sin x)2011dx−∫2010sin2010xdx+C (Applying Integration by parts in the first expression)
=tan x(sin x)2010+∫2010(sin x)2010dx−∫2010sin2010xdx+C
=tan x(sin x)2010+C
∴P(π3)=tan π3=√3