The integral of the sum of two functions is simply the sum of the integrals of each individual function. This is a fundamental property of integration known as the linearity of integration.
Understanding the Concept
Imagine you have two functions, f(x) and g(x). Their sum is represented by f(x) + g(x). The integral of this sum, denoted by:
∫[f(x) + g(x)] dx
is equivalent to:
∫f(x) dx + ∫g(x) dx
This means you can integrate each function separately and then add their individual integrals to obtain the integral of their sum.
Practical Example
Let's say you have the following functions:
- f(x) = x²
- g(x) = 2x
Their sum is f(x) + g(x) = x² + 2x. To find the integral of this sum, you can integrate each function separately:
- ∫x² dx = (x³/3) + C₁
- ∫2x dx = x² + C₂
Where C₁ and C₂ are constants of integration.
Therefore, the integral of the sum of the two functions is:
∫(x² + 2x) dx = (x³/3) + C₁ + x² + C₂
Which can be simplified as:
∫(x² + 2x) dx = (x³/3) + x² + C
Where C = C₁ + C₂ is the combined constant of integration.
Key Points
- The linearity of integration allows us to break down complex integrals into simpler ones.
- This property is extremely useful for solving various integration problems.
- It simplifies the integration process by allowing us to focus on integrating individual functions separately.
