Let `f(x)` be a continuous and periodic function such that `f(x)=f(x+T)` for all `xepsilonR,Tgt0`.If `int_(-2T)^(a+5T)f(x)dx=19(ag0)` and `int_(0)^(T)f(x)dx=2`, then find the value of `int_(0)^(a)f(x)dx`.

Explanation

To solve the problem step by step, we will utilize the properties of periodic functions and definite integrals.

Given:
1. \( f(x) \) is a continuous and periodic function with period \( T \).
2. \( \int_{-2T}^{a + 5T} f(x) \, dx = 19 \) (for \( a > 0 \))
3. \( \int_{0}^{T} f(x) \, dx = 2 \)

We need to find the value of \( \int_{0}^{a} f(x) \, dx \).
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