`int_(-pi)^(pi)(sin^(75)x+x^(125))dx=0`

Prove that : (i) int_(-pi)^(pi) x^(10) sin^(7) x dx =0 (ii) int_(-pi)^(pi) (sin^(25) x+x^(75))dx=0 (iii) int_(-pi)^(pi) e^(|x|) dx= 2 (e-1) (iv) int_(-pi//2)^(pi//2) sin^(9) x dx =0

Prove that : (i) int_(-pi)^(pi) x^(10) sin^(7) x dx =0 (ii) int_(-pi)^(pi) (sin^(25) x+x^(75))dx=0 (iii) int_(-pi)^(pi) e^(|x|) dx= 2 (e-1) (iv) int_(-pi//2)^(pi//2) sin^(9) x dx =0

int_(0)^(pi) sin^(7) x dx=

int_(-pi)^(pi)sin^(2)x.cos^(2)x dx=

int_(-pi)^( pi)(sin^(2)x)/(1+a^(x))dx

If int_(0)^(pi//2) sin^(6) x dx = (pi)/(32) then int_(-pi)^(pi) (sin^(6)x + cos^(6) x)dx=

int_(0)^( pi)sin(2*x)dx

int_(pi)^(10pi) |sin x| dx=

int_(0)^( pi/2)(sin x)*dx