Find `I=int _0^((3pi)/2)cos^4(3x)sin^2(6x)dx`

u,=int_(0)^((pi)/(2))cos((2 pi)/(3)sin^(2)x)dx and v,=int_(0)^((pi)/(2))cos((pi)/(3)sin x)dx

Suppose _((pi)/(2))cos(pi sin^(2)x)dx and I_(2)int_(0)^(3)cos(2 pi sin^(2)x)dx and I_(3)=int_(0)^((pi)/(2))cos(pi sin x)dx, then

int_0^(pi/2) sin^3x/(sin^3x+cos^3x)dx

int_0^(pi/2) sin^3x/(sin^3x+cos^3x)dx

Evaluate: int_0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx

Evaluate: int_0^(pi//2)(cos^2x)/(cos^2x+4sin^2x)dx

Evaluate: int_(0)^((pi)/(2))|(cos^(4)x,7),(sin^(5)x,3)| dx.

(i) int_0^(pi//2) (sin^3x)/(sin^3x+cos^3x) dx (ii) int_0^(pi//2) (cos^3x)/(sin^3x+cos^3x) dx (iii) int_0^(pi//2) (sin^4x)/(sin^4x+cos^4x) dx (iv) int_0^(pi//2) (cos^5x)/(sin^5x+cos^5x) dx (v) int_0^(pi//2) (sin^5x)/(sin^5x+cos^5x) dx

evaluate: int_0^(pi/2) (cos^(3/2)x)/(sin^(3/2)x+cos^(3/2)x)dx