If `f(t)` is an odd function, then prove that `varphi(x)=int_a^xf(t)dt` is an even function.

It is known that f(x) is an odd function and has a period p . Prove that int_(a)^(x)f(t)dt is also periodic function with the same period.

Prove that the derivative of an even function is an odd function and that of an odd function is an even function.

Statement I The function f(x) = int_(0)^(x) sqrt(1+t^(2) dt ) is an odd function and STATEMENT 2 : g(x)=f'(x) is an even function , then f(x) is an odd function.

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an odd function, then

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an even function, then

If f is an integrable function, show that int_(-a)^a xf(x^2)dx=0

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. int_(0)^(1)f(x)dx=

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. Identify the correct statements (s):

The function f(x)=log(x+sqrt(x^(2)+1)) , is (a) an even function (b) an odd function (c ) a periodic function (d) Neither an even nor an odd function.

It is known that f(x) is an odd function in the interval [-p/2, p/2] and has a period p , Prove that int_a^xf(t)dt is also periodic function with the same period.