If f(x)=limy→xsin2y-sin2xy2-x2, then ∫4xf(x)dx=
cos2x+C
2cos2x+C
–cos2x+C
Explanation for correct option:
Find the value of ∫4xf(x)dx:
Given expression,
f(x)=limy→xsin2y-sin2xy2-x2
Since, the function form 00 at y=x, then
Applying L' Hospital rule,
fx=limy→x2sinycosy2y-0=limy→x2siny2ycosy=limy→xsin2y2y=sin2x2x
Now,
∫4xf(x)dx=∫4xsin2x2xdx=2∫sin2xdx=-2cos2x2+C=-cos2x+C
Hence, the correct option is C.
The maximum value of f(x)=sin2x1+cos2xcos2x1+sin2xcos2xcos2xsin2xcos2xsin2x,x∈R is:
Use the factor theorem to determine whether g(x) is a factor of f(x)
f(x)=22x2+5x+2;g(x)=x+2