If `g(x)=int_(0)^(x)cos^(4)t dt`, then prove that `g(x+pi)=g(x)+g(pi)`.

If g(x)=int_(0)^(x)cos^(4) t dt , then (x+pi) equals

If f(x)=int_0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s f(x)+f(pi) (b) f(x)+2(pi) f(x)+f(pi/2) (d) f(x)+2f(pi/2)

Let f(x) be a continuous function AAx in R , except at x=0, such that g(x)= int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx

If g(x)=int_0^x2|t|dt ,then (a) g(x)=x|x| (b) g(x) is monotonic (c) g(x) is differentiable at x=0 (d) gprime(x) is differentiable at x=0

f,g, h , are continuous in [0, a],f(a-x)=f(x),g(a-x)=-g(x),3h(x)-4h(a-x)=5. Then prove that int_0^af(x)g(x)h(x)dx=0

If g is inverse of f then prove that f''(g(x))=-g''(x)(f'(g(x)))^(3).

If f: RrarrR and g:RrarrR are two given functions, then prove that 2min.{f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

Off(x)=int_0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s (a) f(x)+f(pi) (b) f(x)+2(pi) (c) f(x)+f(pi/2) (d) f(x)+2f(pi/2)

By using the properties of definite integrals, evaluate the integrals Show that int_(0)^(a)f(x)g(x)dx=2int_(0)^(a)f(x)dx , if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4 .

Ifg(x)=int_0^x(|sint|+|cost|)dt ,t h e ng(x+(pin)/2) is equal to, where n in N , g(x)+g(pi) (b) g(x)+g((npi)/(n2)) g(x)+g(pi/2) (d) none of these