STATEMENT 1: `int_0^pixsinxcos^2xdx=pi/2int_0^pisinxcos^2x dxdot` STATEMENT 2: `int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dxdot`

"int_(0)^( pi)x sin^(2)xdx

int_0^(pi//2) x^2 cos 2x dx

STATEMENT 1:int_(0)^( pi)sin^(100)x cos^(99)xdx is z ero STATEMENT 2:int_(a)^(b)f(x)dx=int_(a+c)^(b+c)f(x-c)dx, and for odd function,int_(-a)^(a)f(x)dx=0

int_(0)^( pi/2)x sin xdx

int_(0)^( pi)xf(sin x)dx=(pi)/(2)int_(0)^( pi)f(sin x)dx

(i) int_0^(pi/2) sin^2 x dx (ii) int_0^(pi//2) cos^2 x dx

STATEMENT-1 : int_(0)^((pi)/(2))(xsinxcosx)/(sin^(4)x+cos^(4)x)dx=(pi^(2))/(32) and STATEMENT-2 : int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx

If f(a+b-x0=f(x) , then prove that int_a^b xf(x)dx=((a+b)/2)int_a^bf(x)dxdot

If f(a+b-x)=f(x),\ then prove that \ int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dx

STATEMENT-1 : int_(-2pi)^(5pi)cot^(-1)(tanx)dx=7(pi^(2))/(2) and STATEMENT-2 : int_(a)^(b)f(x)dx=int_(a)^(c )f(x)dx+int_(c )^(b)f(x)dx , a lt c lt b