Calculate the intergral, assuming that internal 5,0 f(x)dx=-7 and internal 5,0 g(x)dx=35. What is intergal 5,0 (f(x)+g(x))dx

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.

Calculate the integral, assuming that integral from 0 to 5 of f(x) dx = -7 and integral from 0 to 5 of g(x) dx = 35. What is the integral from 0 to 5 of (f(x) + g(x)) dx?

Learn how to solve the integral of f(x) + g(x) over the interval [0, 5] using the properties of definite integrals. With given values for the integrals of f(x) and g(x), this solution demonstrates step-by-step application of the linearity property. Suitable for students in Grades 11-12 or early college.

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