Let's dive into solving the integral of sin(2x)cos(2x)sin(x)cos(x). This looks complex, but we can simplify it using trigonometric identities. Buckle up, guys, because we're about to untangle this trigonometric knot!

Understanding the Integral

Before we even think about integrating, let's rewrite the expression to make it easier to work with. Our integral is:

∫ sin(2x)cos(2x)sin(x)cos(x) dx

The key here is to use trigonometric identities to simplify the expression inside the integral. We'll start with the double-angle identities. Remember that:

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) - sin²(x)

Substituting the first identity into our integral, we get:

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

Which simplifies to:

∫ 2sin²(x)cos²(x)cos(2x) dx

Now, let's use another identity. Recall that sin(2x) = 2sin(x)cos(x). Therefore, sin(x)cos(x) = sin(2x)/2. Substituting this into our integral:

∫ 2(sin(2x)/2)²cos(2x) dx

Which simplifies to:

∫ 2(sin²(2x)/4)cos(2x) dx

Further simplifying, we have:

∫ (1/2)sin²(2x)cos(2x) dx

Now, this looks much more manageable!

Applying U-Substitution

To solve this integral, we’ll use u-substitution. Let's set:

u = sin(2x)

Then, the derivative of u with respect to x is:

du/dx = 2cos(2x)

So, du = 2cos(2x) dx, which means cos(2x) dx = du/2. Substituting these into our integral, we get:

∫ (1/2)u² (du/2)

Which simplifies to:

(1/4) ∫ u² du

Integrating and Back-Substitution

Now we can easily integrate u² with respect to u:

(1/4) * (u³/3) + C

Which is:

(1/12)u³ + C

Now, substitute back u = sin(2x):

(1/12)sin³(2x) + C

So, the final answer is:

∫ sin(2x)cos(2x)sin(x)cos(x) dx = (1/12)sin³(2x) + C

Breaking Down the Steps Further

Initial Simplification

First, let's recap the initial simplification. The most important step is recognizing the double angle identities. When you see sin(2x) and cos(2x), think of ways to express them in terms of sin(x) and cos(x). This allows you to consolidate terms and potentially find cancellations or simplifications. The original integral looks daunting, but breaking it down using these identities makes it solvable. Remember, practice makes perfect, so try different combinations to see what works best!

The Power of Substitution

The u-substitution technique is powerful, especially when you notice a function and its derivative (or a multiple thereof) within the integral. In our case, recognizing that the derivative of sin(2x) is related to cos(2x) was key. The goal is to transform the integral into a simpler form that you can easily integrate using standard rules. Always remember to substitute back to the original variable after integrating!

Alternative Approaches

While u-substitution worked well here, there might be other ways to tackle this integral. For instance, you could try using power-reducing formulas for sin²(2x) to express it in terms of cos(4x). This might lead to a different, but equivalent, form of the integral. Exploring different approaches can give you a deeper understanding of the problem and improve your problem-solving skills.

Common Mistakes to Avoid

• Forgetting the Constant of Integration: Always add "+ C" after integrating. It represents the family of antiderivatives.
• Incorrectly Applying Identities: Double-check your trigonometric identities before using them. A small mistake can lead to a completely wrong answer.
• Not Substituting Back: After u-substitution, don't forget to substitute back to the original variable.

Conclusion

So, there you have it! The integral of sin(2x)cos(2x)sin(x)cos(x) is (1/12)sin³(2x) + C. By using trigonometric identities and u-substitution, we turned a seemingly complex problem into a manageable one. Keep practicing, and you'll become a master of integration in no time! Remember, the journey of a thousand miles begins with a single step, and every integral you solve brings you closer to mathematical enlightenment. You got this, guys! Keep on integrating!

Now, let's try another integral problem to solidify your understanding. Think about the integral of cos(x)sin²(x). Can you see how to approach it? Hint: u-substitution might be your friend again! Go forth and conquer, my mathematical comrades!

Additional Tips and Tricks

When dealing with trigonometric integrals, remember these handy tips:

1. Know Your Identities: A strong grasp of trigonometric identities is crucial. Keep a list handy and refer to it often.
2. Look for Patterns: Try to identify patterns that suggest a particular substitution or simplification technique.
3. Don't Be Afraid to Experiment: If one approach doesn't work, try another. Sometimes it takes a bit of trial and error to find the best solution.
4. Simplify First: Always simplify the integrand as much as possible before attempting to integrate.

By following these tips, you'll be well on your way to mastering trigonometric integrals. Now, let's explore some more complex examples to further challenge your skills.

Advanced Examples

Consider the integral of sin³(x)cos²(x). This one requires a bit more manipulation. First, we can rewrite sin³(x) as sin²(x)sin(x):

∫ sin²(x)sin(x)cos²(x) dx

Then, use the identity sin²(x) = 1 - cos²(x):

∫ (1 - cos²(x))sin(x)cos²(x) dx

Now, let u = cos(x), so du = -sin(x) dx. Substituting, we get:

∫ (1 - u²)u² (-du)

Which simplifies to:

-∫ (u² - u⁴) du

Integrating, we have:

-(u³/3 - u⁵/5) + C

Substituting back u = cos(x):

-(cos³(x)/3 - cos⁵(x)/5) + C

So, the final answer is:

-cos³(x)/3 + cos⁵(x)/5 + C

This example demonstrates how to handle odd powers of sine and cosine. The key is to isolate one sine or cosine term and use the Pythagorean identity to rewrite the remaining terms in terms of the other trigonometric function.

Another challenging example is the integral of tan(x). This can be rewritten as:

∫ sin(x)/cos(x) dx

Let u = cos(x), so du = -sin(x) dx. Substituting, we get:

∫ -du/u

Integrating, we have:

-ln|u| + C

Substituting back u = cos(x):

-ln|cos(x)| + C

So, the final answer is:

-ln|cos(x)| + C

These examples illustrate the versatility of trigonometric identities and substitution techniques. By mastering these tools, you'll be able to tackle a wide range of trigonometric integrals.

Final Thoughts

Integrating trigonometric functions can be challenging, but with a solid understanding of trigonometric identities and substitution techniques, you can conquer even the most complex integrals. Remember to practice regularly, review your identities, and don't be afraid to experiment with different approaches. Keep pushing your boundaries, and you'll become a true master of integration. Happy integrating, everyone!