Exploring the Fundamental Concept of Integration | Evaluating the Sum of Two Functions Using the Integral Symbol (∫)

a∫b [f(x) + g(x)] dx

The expression “∫” denotes the integral symbol, representing the process of integration

The expression “∫” denotes the integral symbol, representing the process of integration. The integral of a function is a fundamental concept in calculus, used to determine the area under a curve or to evaluate the total accumulation of a quantity.

In the given expression, “a” and “b” represent the limits of integration, specifying the interval over which the integration is performed. These limits dictate the range of values for which the function is integrated.

The term “[f(x) + g(x)]” represents the sum of two functions, f(x) and g(x), enclosed in square brackets. The integral of the sum of two functions is equal to the sum of the integrals of each individual function. Therefore, the expression “∫ [f(x) + g(x)] dx” can be rewritten as ∫f(x) dx + ∫g(x) dx.

To evaluate the given integral, you can separately integrate each function and then add their results together.

So, express the integral as:

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Then, find the antiderivative (the indefinite integral) of f(x) and g(x), which involves reversing the process of differentiation. Essentially, you need to find functions F(x) and G(x) that, when derived, yield f(x) and g(x), respectively.

Once you have the antiderivatives F(x) and G(x), evaluate them at the upper limit b and lower limit a. Then subtract the result of F(b) – F(a) and G(b) – G(a) to find the definite integral for each function.

Finally, add the individual integrals together:

∫a∫b [f(x) + g(x)] dx = ∫a∫b f(x) dx + ∫a∫b g(x) dx

Note: Make sure to apply any additional rules or theorems that may pertain to the functions f(x) and g(x) or the specific case you are examining, such as the linearity of integration or the substitution rule. These may affect the way you evaluate the integral.

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