Hey guys! Let's dive into solving a fun integral problem today. We're going to tackle the integral of sin(2x)cos(2x)sin(x)cos(x). This might look intimidating at first glance, but don't worry, we'll break it down step-by-step. By the end of this guide, you'll not only know how to solve this specific integral but also gain some valuable insights into trigonometric integrals in general.

Understanding the Basics

Before we jump into the nitty-gritty, let's refresh some fundamental trigonometric identities. These identities are the bread and butter of solving trigonometric integrals, and you'll find them super handy. Key identities include:

• Double Angle Formulas:
  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos²(x) - sin²(x)
• Pythagorean Identity:
  • sin²(x) + cos²(x) = 1

Knowing these identities will allow us to simplify complex expressions into manageable forms. For example, recognizing that sin(2x) is simply 2sin(x)cos(x) is crucial for our problem. These aren't just formulas to memorize; they're tools to simplify and manipulate expressions, making integration much easier. Think of them as your best friends when you're navigating the world of trigonometric integrals. Understanding how to use them effectively can transform seemingly impossible integrals into straightforward problems. So, keep these identities close, and you'll be well-equipped to handle a wide variety of trigonometric challenges. Let's keep these in mind as we move forward; they'll be super useful in making our lives easier.

Step-by-Step Solution

Okay, let's get our hands dirty and solve the integral step by step. Our goal is to evaluate:

∫ sin(2x)cos(2x)sin(x)cos(x) dx

Step 1: Simplify Using the Double Angle Formula

The first thing we notice is the sin(2x) term. We can use the double angle formula to rewrite it as 2sin(x)cos(x). So our integral becomes:

∫ (2sin(x)cos(x))cos(2x)sin(x)cos(x) dx

This simplifies to:

∫ 2sin²(x)cos²(x)cos(2x) dx

Step 2: Further Simplification

Now, notice that 2sin²(x)cos²(x) can be written as ½(4sin²(x)cos²(x)), and 4sin²(x)cos²(x) is just (2sin(x)cos(x))². Using the double angle formula again, we get (sin(2x))². Thus, our integral transforms to:

∫ ½ sin²(2x) cos(2x) dx

Step 3: U-Substitution

This is where the magic happens! Let's use u-substitution to make this integral even easier. Let u = sin(2x), so du = 2cos(2x) dx. This means cos(2x) dx = ½ du. Substituting these into our integral, we get:

∫ ½ u² (½ du) = ¼ ∫ u² du

Step 4: Integrate

Now we have a simple power rule integral. The integral of u² is (u³/3). So, we have:

¼ (u³/3) + C = u³/12 + C

Step 5: Substitute Back

Finally, we need to substitute back to get our answer in terms of x. Since u = sin(2x), we have:

sin³(2x) / 12 + C

So, the solution to the integral ∫ sin(2x)cos(2x)sin(x)cos(x) dx is sin³(2x) / 12 + C. Congrats, you've solved it!

Alternative Approach

There's always more than one way to skin a cat, right? Let's explore an alternative approach to solving this integral. This method might give you a different perspective and reinforce your understanding.

Step 1: Rewrite Using Double Angle Formulas

Start with the original integral:

∫ sin(2x)cos(2x)sin(x)cos(x) dx

We already know that sin(2x) = 2sin(x)cos(x). So, let's substitute that in:

∫ (2sin(x)cos(x))cos(2x)sin(x)cos(x) dx

This simplifies to:

∫ 2sin²(x)cos²(x)cos(2x) dx

Step 2: Use Another Double Angle Formula

Recall that sin(2x) = 2sin(x)cos(x). Therefore, sin²(2x) = 4sin²(x)cos²(x). We can rewrite our integral using this identity:

∫ ½ sin²(2x) cos(2x) dx

Step 3: Apply Another Trig Identity

We can use the identity sin²(θ) = ½ [1 - cos(2θ)]. Applying this to sin²(2x), we get:

sin²(2x) = ½ [1 - cos(4x)]

Substitute this back into the integral:

∫ ½ * ½ [1 - cos(4x)] cos(2x) dx = ¼ ∫ [cos(2x) - cos(4x)cos(2x)] dx

Step 4: Product-to-Sum Formula

Now we need to deal with the cos(4x)cos(2x) term. We can use the product-to-sum formula:

cos(A)cos(B) = ½ [cos(A + B) + cos(A - B)]

Applying this, we get:

cos(4x)cos(2x) = ½ [cos(6x) + cos(2x)]

Substitute this back into the integral:

¼ ∫ [cos(2x) - ½ [cos(6x) + cos(2x)]] dx

Step 5: Simplify and Integrate

Simplify the integral:

¼ ∫ [½ cos(2x) - ½ cos(6x)] dx = ⅛ ∫ [cos(2x) - cos(6x)] dx

Now integrate each term:

⅛ [½ sin(2x) - ⅙ sin(6x)] + C = 1/16 sin(2x) - 1/48 sin(6x) + C

So, the alternative solution to the integral ∫ sin(2x)cos(2x)sin(x)cos(x) dx is 1/16 sin(2x) - 1/48 sin(6x) + C. This might look different from our first answer, but remember, trigonometric expressions can often be written in multiple equivalent forms.

Key Takeaways

• Master Trigonometric Identities: Knowing your trig identities is crucial. They allow you to simplify complex integrals into manageable forms.
• U-Substitution is Your Friend: U-substitution can transform tricky integrals into simpler ones.
• Multiple Approaches: There's often more than one way to solve an integral. Exploring different methods can deepen your understanding.
• Don't Be Afraid to Simplify: Always look for opportunities to simplify the integral before you start integrating.

Practice Problems

To solidify your understanding, try solving these similar integrals:

1. ∫ sin(x)cos(x) dx
2. ∫ sin³(x)cos(x) dx
3. ∫ cos³(x)sin(x) dx

Solving these problems will help you become more comfortable with trigonometric integrals and the techniques we've discussed.

Conclusion

Alright, we've successfully tackled the integral of sin(2x)cos(2x)sin(x)cos(x) using a couple of different methods. Remember, practice makes perfect, so keep working on those integrals, and you'll become a pro in no time! Understanding trigonometric identities and mastering techniques like u-substitution are key to solving these types of problems. So keep those identities handy, and don't hesitate to explore different approaches. Happy integrating, and keep an eye out for more math adventures!