if f(x) = g(x) + 7 for 3 <= x <= 5 , then integrate [f(x) + g(x)] dx from 3 to 5 = ?

Answer 1

To find the integral of [f(x) + g(x)] over the interval from 3 to 5, we need to first find the integral of f(x) + g(x).

Since f(x) = g(x) + 7, we can substitute this into the integral:

∫[f(x) + g(x)] dx from 3 to 5 = ∫[(g(x) + 7) + g(x)] dx from 3 to 5.

Expanding the expression inside the integral:

= ∫[2g(x) + 7] dx from 3 to 5.

Now, we can integrate term by term:

= ∫2g(x) dx + ∫7 dx from 3 to 5.

The integral of 2g(x) will depend on the specific function g(x) and cannot be determined without knowing its form.

However, the integral of the constant term 7 is simply 7x:

= 2∫g(x) dx + 7x from 3 to 5.

Now, if you know the specific form of g(x), you can determine the integral of g(x) and substitute the limits of integration from 3 to 5 to evaluate the integral.