Hey guys! Today, we're diving into a fun calculus problem: integrating sin(2x)cos(2x)sin(x)cos(x). This looks a bit intimidating at first, but don't worry, we'll break it down step by step. By using trigonometric identities and a little bit of algebraic manipulation, we can simplify this integral into something much more manageable. So, grab your pencils, and let's get started!

Understanding the Integral

Before we jump into the solution, let's make sure we understand what we're trying to achieve. The goal is to find the indefinite integral of the function f(x) = sin(2x)cos(2x)sin(x)cos(x). This means we want to find a function F(x) such that when we differentiate F(x), we get back our original function f(x). Integrals can be used in various applications, from physics to engineering, to calculate areas, volumes, and other important quantities.

Trigonometric Identities to the Rescue

The key to solving this integral lies in using trigonometric identities. These identities will help us simplify the expression inside the integral. The main identities we'll use are:

1. Double Angle Identity: sin(2x) = 2sin(x)cos(x)
2. Another Double Angle Identity: sin(4x) = 2sin(2x)cos(2x)

These identities allow us to rewrite the original expression in a more simplified form, making it easier to integrate. Remember, the goal is to reduce the complexity of the integrand so that we can apply standard integration techniques.

Step-by-Step Solution

Let's walk through the solution step by step. This will help you understand the process and see how each step builds upon the previous one.

Step 1: Rewrite the Integral

Our integral is ∫sin(2x)cos(2x)sin(x)cos(x) dx. We can start by applying the double angle identity sin(2x) = 2sin(x)cos(x) to the sin(x)cos(x) term. This gives us:

∫sin(2x)cos(2x) * (1/2)sin(2x) dx = (1/2)∫sin²(2x)cos(2x) dx

Step 2: Apply Another Double Angle Identity

Now, we have (1/2)∫sin²(2x)cos(2x) dx. Notice that sin(4x) = 2sin(2x)cos(2x). We can rewrite sin²(2x)cos(2x) as (1/2)sin(2x) * 2sin(2x)cos(2x) = (1/2)sin(2x)sin(4x).

So our integral becomes:

(1/2)∫(1/2)sin(2x)sin(4x) dx = (1/4)∫sin(2x)sin(4x) dx

Step 3: Product-to-Sum Identity

To simplify further, we can use the product-to-sum identity:

sin(A)sin(B) = (1/2)[cos(A - B) - cos(A + B)]

In our case, A = 2x and B = 4x. Applying the identity, we get:

sin(2x)sin(4x) = (1/2)[cos(2x - 4x) - cos(2x + 4x)] = (1/2)[cos(-2x) - cos(6x)]

Since cos(-x) = cos(x), this simplifies to:

(1/2)[cos(2x) - cos(6x)]

So our integral becomes:

(1/4)∫(1/2)[cos(2x) - cos(6x)] dx = (1/8)∫[cos(2x) - cos(6x)] dx

Step 4: Integrate

Now we can integrate each term separately:

(1/8)∫cos(2x) dx - (1/8)∫cos(6x) dx

The integral of cos(ax) is (1/a)sin(ax). Therefore:

(1/8) * (1/2)sin(2x) - (1/8) * (1/6)sin(6x) + C

Simplifying, we get:

(1/16)sin(2x) - (1/48)sin(6x) + C

So, the integral of sin(2x)cos(2x)sin(x)cos(x) is (1/16)sin(2x) - (1/48)sin(6x) + C, where C is the constant of integration.

Common Mistakes to Avoid

When solving integrals like this, there are a few common mistakes to watch out for:

• Forgetting the Constant of Integration: Always remember to add the constant of integration, C, when finding indefinite integrals. This represents the family of functions that have the same derivative.
• Incorrectly Applying Trigonometric Identities: Make sure you are using the identities correctly. A small mistake here can lead to a completely wrong answer.
• Not Simplifying Enough: Sometimes, you might stop simplifying too early. Always try to simplify the integrand as much as possible before integrating.
• Sign Errors: Pay close attention to signs, especially when dealing with trigonometric functions.

Alternative Approaches

While we solved this integral using trigonometric identities, there might be other approaches you could take. For example, you could try using substitution. However, in this case, trigonometric identities provide a more straightforward solution.

Using Substitution (Less Efficient)

Let's explore how substitution could be applied, though it's less efficient here to illustrate the concept.

Starting with: (1/2)∫sin²(2x)cos(2x) dx

Let u = sin(2x), then du = 2cos(2x) dx, so cos(2x) dx = (1/2) du

The integral becomes:

(1/2)∫u² * (1/2) du = (1/4)∫u² du

Integrating:

(1/4) * (u³/3) + C = (1/12)u³ + C

Substituting back:

(1/12)sin³(2x) + C

While this result is correct, it looks different from our previous answer. This highlights an important point: there can be multiple equivalent forms of the same integral. To show they are equivalent, you would need to use trigonometric identities to transform one into the other. For example, expanding sin³(2x) and using further identities would eventually lead to the same expression we found earlier.

Practical Applications

Understanding how to solve integrals like this isn't just a theoretical exercise. These skills are essential in various fields. For example:

• Physics: Calculating the motion of objects, wave behavior, and electromagnetic fields.
• Engineering: Designing structures, analyzing circuits, and modeling systems.
• Computer Graphics: Creating realistic lighting and shading effects.
• Data Analysis: Modeling and predicting trends in data.

Practice Problems

To solidify your understanding, try solving these similar integrals:

1. ∫cos(2x)cos(4x) dx
2. ∫sin(3x)cos(5x) dx
3. ∫sin²(x)cos(x) dx

Work through these problems step by step, using the techniques we discussed. Don't be afraid to make mistakes – that's how you learn!

Conclusion

Integrating sin(2x)cos(2x)sin(x)cos(x) might have seemed daunting at first, but by using trigonometric identities and breaking it down into smaller steps, we were able to find the solution. Remember to always simplify your expressions as much as possible and watch out for common mistakes. With practice, you'll become more comfortable with these types of integrals.

Keep practicing, and you'll master these techniques in no time! Happy integrating, guys! Hope this helps you and good luck!