`cos^4x+sin^4x=?`

If f(x) = |[cos^4x, 2 sin^4x, 3 sin^2 2x], [3 2 cos^4 x, 2 sin^4 x , sin^2 2x],[2 cos^4x, 3 2 sin^4x, sin^2 2x]| then frac{1}{5} f'(0) is equal to

int cos2(x)/(cos^(4)x+sin^(4)x)dx

int_(0)^((pi)/(2))(x sin x cos x)/(cos^(4)x+sin^(4)x)dx=

int_(0)^(pi)(cos^(4)x-sin^(4)x)dx=

int(cos^(4)x-sin^(4)x)dx

If A=[[2,3,0],[1,sin^(2)x,-cos^(2)x],[cos^(2)x,cos^(4)x,sin^(4)x]], then det "[" adj "{" adj "{" adj "......{" adj "A}}] ,(adjoint is taken r times)

Evaluate int_(0)^(pi//2)(xsinxcosx)/(cos^(4)x+sin^(4)x)dx

The minimum value of |sinx+cosx+(cosx+sinx)/(cos^(4)x-sin^(4)x)| is