Given a function `f(x)` such that It is integrable over every interval on the real line, and `f(t+x)=f(x),` for every `x` and a real `tdot` Then show that the integral `int_a^(a+t)f(x)dx` is independent of `adot`

If f(x) is even then int_(-a)^(a)f(x)dx ….

The value of the integral int_0^(2a)[(f(x))/({f(x)+f(2a-x)})]dx "is equal to "a

If f(x)=f(a+x) then show that int_(0)^(2a)f(x)dx=2int_(0)^(a)f(x)dx .

If f(x)=x+int_0^1 t(x+t) f(t)dt, then find the value of the definite integral int_0^1 f(x)dx.

A periodic function with period 1 is integrable over any finite interval. Also, for two real numbers a,b and two unequal non-zero positive integers m and n int_a^(a+n) f(x)dx=int_b^(b+m) f(x) calculate the value of int_m^n f(x) dx

Let f(x) = x-[x] , for every real number x, where [x] is integral part of x. Then int_(-1) ^1 f(x) dx is

If f(x) is continuous and int_(0)^(9)f(x)dx=4 , then the value of the integral int_(0)^(3)x.f(x^(2))dx is

A function f(x) is such that f(x)=0 has 8 distinct real roots and f(4+x)=f(4-x) for x in R . Sum of real roots of the equation f(x)=0 is

If f(x) is an odd function then int_(-a)^(a)f(x)dx is …………. .

A function f, continuous on the positive real axis, has the property that for all choices of x gt 0 and y gt 0, the integral int_(x)^(xy)f(t)dt is independent of x (and therefore depends only on y). If f(2) = 2, then int_(1)^(e)f(t)dt is equal to