Given the integrals:

$\int_0^5 f(x)\,dx = -7 \quad \text{and} \quad \int_0^5 g(x)\,dx = 35$

We want to calculate the integral:

$\int_0^5 (f(x) + g(x))\,dx$

By the property of integrals, the integral of the sum of two functions is equal to the sum of the integrals of the functions:

$\int_0^5 (f(x) + g(x))\,dx = \int_0^5 f(x)\,dx + \int_0^5 g(x)\,dx$

Substituting the given values:

$\int_0^5 (f(x) + g(x))\,dx = -7 + 35 = 28$

So, the value of the integral is $28$.

Would you like further details or have any other questions?

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Here are 5 related questions:

1. What are the properties of definite integrals?
2. How do you interpret the result of an integral geometrically?
3. Can the integral of a function be negative? What does it represent?
4. How would you find the integral if $f(x)$ and $g(x)$ were given as explicit functions?
5. What if the limits of integration were different, say from 2 to 5 instead of 0 to 5?

Tip: The sum and difference properties of integrals are very useful when dealing with multiple functions.