Question

Suppose that ₅∫⁰f(x) dx = 5 and ₅∫⁰g(x) dx = 12, calculate the following integrals.

(a) ₅∫⁰(f(x) + g(x)) dx

Answer 1

The integral of 5∫0(f(x) + g(x)) dx equals 17, based on the given integrals 5∫0f(x) dx = 5 and 5∫0g(x) dx = 12, demonstrating the additive property of integrals.

Step-by-step explanation:

When solving for the integral 5∫0(f(x) + g(x)) dx, we can use the linearity of integration to split the integral into the sum of two separate integrals:

1. 5∫0f(x) dx
2. 5∫0g(x) dx

Because we have been given that 5∫0f(x) dx = 5 and 5∫0g(x) dx = 12, we simply add these two values together to get the result.

The integral of the sum is thus 5 + 12 = 17. So, the integral 5∫0(f(x) + g(x)) dx equals 17.

This result showcases an important rule in calculus known as the additive property of integrals, which states that the integral of a sum of functions is equal to the sum of the integrals of each function provided the integrals exist.