Suppose that 𝑓 and 𝑔 are continuous functions and that 𝑓 is an odd function, i.e. 𝑓(−𝑥)=−𝑓(𝑥) for all 𝑥.Find ∫25(𝑓(𝑥)+5𝑔(𝑥))𝑑𝑥 given that∫−52𝑓(𝑥) 𝑑𝑥=9 and ∫25𝑔(𝑥) 𝑑𝑥=-2.

To solve this problem, we need to use the properties of integrals and the given information.

The integral of a sum of functions is the sum of the integrals of the functions. So, we can write the integral of the function 25(f(x) + 5g(x)) as the sum of the integral of 25f(x) and the integral of 125g(x).

∫25(f(x) + 5g(x)) dx = ∫25f(x) dx + ∫125g(x) dx

We also know that the integral of a constant times a function is the constant times the integral of the function. So, we can write the integral of 25f(x) as 25 times the integral of f(x), and the integral of 125g(x) as 125 times the integral of g(x).

∫25(f(x) + 5g(x)) dx = 25∫f(x) dx + 125∫g(x) dx

We are given that ∫f(x) dx from -5 to 2 is 9 and ∫g(x) dx from 2 to 5 is -2. However, we need to find the integral of f(x) from 2 to 5. Since f is an odd function, the integral of f(x) from -a to a is 0 for any a. Therefore, the integral of f(x) from -5 to 5 is 0, and the integral of f(x) from 2 to 5 is - the integral of f(x) from -5 to 2, which is -9.

So, we can substitute these values into the equation:

∫25(f(x) + 5g(x)) dx = 25*(-9) + 125*(-2) = -225 - 250 = -475.