How can you integrate a formula? Explain it in detail along with mathematical expressions and equations.

Answer 1

Integration is the process of finding the antiderivative of a function. It involves reversing the process of differentiation.

To integrate a formula, we perform the following steps:

Step 1: Identify the function to integrate. Let's say we have a function f(x).

Step 2: Identify the variable of integration. Generally, this will be the variable with respect to which we want to integrate the function. In our case, let's assume it is x.

Step 3: Calculate the antiderivative (also known as the indefinite integral) of the function. This is denoted as F(x). The antiderivative represents a family of functions that, when differentiated, yield the original function f(x). We can symbolically write the antiderivative as ∫f(x)dx = F(x) + C, where C is the constant of integration.

Step 4: Determine the limits of integration if evaluating a definite integral. Definite integration involves finding the area under the curve of the function between two specified limits. The limits are denoted as a and b. The definite integral is written as ∫[a to b]f(x)dx.

Now, let's illustrate the process with an example.

Example: Integrate the function f(x) = 2x.

Step 1: The function to integrate is f(x) = 2x.

Step 2: The variable of integration is x.

Step 3: To find the antiderivative, we integrate with respect to x:

∫2x dx = x^2 + C,

where C is the constant of integration.

Step 4: If we want to evaluate the definite integral of f(x) between 1 and 3, we write:

∫[1 to 3]2x dx.

We substitute the values into the antiderivative:

[ x^2 + C ] evaluated from 1 to 3.

Evaluating at x = 3:

(3^2 + C) = 9 + C.

Evaluating at x = 1:

(1^2 + C) = 1 + C.

To find the definite integral, we subtract the lower limit from the upper limit:

(9 + C) - (1 + C) = 8.

Hence, the definite integral of 2x from 1 to 3 is 8.

This is a basic example of integrating a simple function. In practice, integration involves various techniques such as substitution, integration by parts, trigonometric substitution, etc., depending on the complexity of the function.