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Definite integrals are used to compute the net area under a curve over a specific interval. Unlike indefinite integrals, which represent a family of functions, definite integrals provide a numerical value. They are expressed as \( \int_{a}^{b} f(x) dx \).

To calculate a definite integral, you need to:

• Determine the antiderivative of the integrand.
• Evaluate this antiderivative at the upper limit of integration.
• Subtract the value of the antiderivative at the lower limit from its value at the upper limit.

This process gives you the accumulated value of the function between the specified limits. Understanding this helps you solve real-world problems, such as finding the total distance traveled over a time interval.

The substitution method is a powerful technique for evaluating integrals, especially when the integrand is a composite function. This technique involves finding a substitution that simplifies the integral. Let's break down the steps:

1. **Choose a substitution**: Identify an inner function in the integrand and set it equal to a new variable, often denoted as \( u \).
2. **Differentiate the substitution**: Find the differential of the new variable. This step usually involves calculating \( du \).
3. **Rewrite the integral**: Replace the original variable and differential with the new variable and its differential, converting the integral into a simpler form.
4. **Integrate**: Perform the integration with the new variable.
5. **Substitute back**: Replace the new variable with the original variable to express the final answer in terms of the original variable.

Using substitution helps transform a complex integral into a simpler one, making it easier to solve.

Differentiation and integration are inverse processes in calculus. When using the substitution method, differentiation plays a crucial role. Here's why:

• **Finding the derivative**: To rewrite the integral in terms of a new variable, you first need to differentiate the chosen substitution. This step allows you to express the differential \( dx \) in terms of \( du \).
• **Chain Rule**: Differentiation involves the chain rule, where you calculate the derivative of a composite function.

In the context of our example exercise:

• You start by letting \( u = \cos 2x \).
• Next, you differentiate to find \( du = -2 \sin 2x dx \).
• This differential is essential for rewriting the original integral in terms of \( u \).

Differentiation simplifies the expression, making it possible to convert a complex integral into a more manageable form.