`int_0^9 \ f(x) \ dx,` where `f(x)={(sinx, if 0<=x < pi/2), (1, if pi/2 <= x < 3), (e^(x-3), if 3<=x<9):}`

int_0^a f(a-x) dx=

Evaluate int_0^3 f(x)dx , Where f(x)={(x,+,3,,,0lexle2),(3x,,,,,2lexle3):}

Show that f (x) f (y) = f(x+y) , where f(x)=[(cosx,-sinx,0),(sinx,cosx,0),(0,0,1)] .

Prove that int_(0)^(2a) f(x) dx = 2int_(0)^(a) f(x) dx when f(2a -x) = f(x) and hence evaluate int_(0)^(pi) |cos x| dx .

Evaluate int_0^1e^x{f(x)+f'(x)}dx where f (1)=f(0)=1

Statement-1: int_(0)^(npi+v)|sin x|dx=2n+1-cos v where n in N and 0 le v lt pi . Stetement-2: If f(x) is a periodic function with period T, then (i) int_(0)^(nT) f(x)dx=n int_(0)^(T) f(x)dx , where n in N and (ii) int_(nT)^(nT+a) f(x)dx=int_(0)^(a) f(x) dx , where n in N

Statement-1: int_(0)^(npi+v)|sin x|dx=2n+1-cos v where n in N and 0 le v lt pi . Stetement-2: If f(x) is a periodic function with period T, then (i) int_(0)^(nT) f(x)dx=n int_(0)^(T) f(x)dx , where n in N and (ii) int_(nT)^(nt+a) f(x)dx=int_(0)^(a) f(x) dx , where n in N

Show that int_0^a f(x)g(x)dx = 2 int_0^a f(x)dx , if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4

Property 11: int_0 ^(nT) f(x)= n int_0 ^T f(x) dx where T is the period of the function and n is integer