Hey guys! Let's dive into a classic calculus problem: how to solve the integral of sin³(x)cos³(x). This isn't just some random integral; it's a great example of how to use trigonometric identities and clever substitutions to crack a seemingly tricky problem. We'll break it down into easy-to-follow steps, so even if you're not a math whiz, you'll be able to understand the process. Ready to get started?

We're going to explore this integral comprehensively, from the initial setup to the final answer. The key here is to simplify the expression using trigonometric identities and a well-chosen substitution. By the end, you'll have a clear grasp of not just this integral, but also the general approach to solving similar problems. So, buckle up; it's time to flex those math muscles!

Understanding the Problem: The Integral of sin³(x)cos³(x)

Alright, first things first: let's get a handle on what we're actually dealing with. We're trying to find the integral of sin³(x)cos³(x) with respect to x. This means we want to find a function whose derivative is sin³(x)cos³(x). Seems straightforward, right? Well, it's not as simple as it looks at first glance! The presence of the powers of sine and cosine makes it a bit of a challenge. Our goal is to transform this integral into a form we can easily solve. That's where our toolbox of trigonometric identities and integration techniques comes in handy. Think of it like this: we have a puzzle, and our goal is to find the right pieces and put them together correctly.

The initial look can be intimidating, but trust me, with the right approach, it becomes manageable. The core idea here involves manipulating the integrand (the function inside the integral) to make it easier to integrate. This often means using trigonometric identities to rewrite the expression in a more convenient form. We're going to use a strategic approach. We're going to use trig identities to simplify the integrand. We will then use substitution to further simplify the integral.

Breaking Down the Integral

To make things easier, we'll start by rewriting the integral. The trick is to split off one power of sin(x) and pair it with dx. This is a common strategy when dealing with integrals of this type. It'll become clearer as we go. We can rewrite the original integral ∫sin³(x)cos³(x)dx as ∫sin(x)sin²(x)cos³(x)dx. This manipulation doesn't change the value of the integral; it just sets us up for the next step. Notice that we've isolated a single sin(x) term. This might seem like a small change, but it's going to be crucial for our substitution. Think of it as preparing the ground for the transformation that's about to happen. This rearrangement is a strategic setup, preparing us for the next, more significant steps.

Now, let's bring in a key trigonometric identity. We know that sin²(x) + cos²(x) = 1. This is our magic wand! We can use this identity to rewrite sin²(x) in terms of cos²(x). Specifically, sin²(x) = 1 - cos²(x). This substitution is the heart of the simplification process. It's what allows us to change our integral into a more manageable form. By replacing sin²(x), we’re essentially changing the “language” of the integral to make it easier to “understand” and solve. This is the crucial step in our approach to solve the integral.

Applying Substitution: The Key to Simplification

Okay, now that we've set the stage, it's time for the main act: substitution. This is where things really start to click! We're going to choose a smart substitution to simplify the integral even further. The main idea is to replace a part of the integrand (the function being integrated) with a new variable, making the expression simpler to handle.

Let's choose u = cos(x). The reason we pick this is because the derivative of cos(x), which is -sin(x)dx, is also present (or can be easily created) in our integral. This allows us to convert the entire integral into a function of u. So, if u = cos(x), then du = -sin(x)dx. Notice that the negative sign is important here; we'll need to account for it later. This is a common and powerful technique in integral calculus, enabling us to transform complex integrals into simpler ones.

Transforming the Integral with Substitution

Now that we have our substitution, let's rewrite the integral in terms of u. Remember our rewritten integral: ∫sin(x)sin²(x)cos³(x)dx. Using our substitutions u = cos(x) and sin²(x) = 1 - cos²(x), and knowing that du = -sin(x)dx, we can transform the integral. The integral now becomes ∫(1 - u²)u³(-du). The sin(x)dx term has been replaced by -du, cos³(x) has become u³, and sin²(x) has been replaced by (1 - u²). The negative sign comes from the du term, which we pull out front, giving us -∫(1 - u²)u³du. Notice how the variable x has completely disappeared; we now only have u to work with. This is the beauty of substitution: we've changed the problem into something that looks and feels easier to solve.

Next, let's simplify the integrand. We can distribute the u³ term and rewrite the integral as -∫(u³ - u⁵)du. Now, our integral is a simple polynomial, which is incredibly easy to integrate. This step might seem simple, but it's crucial for getting to the final answer. This distribution step sets us up perfectly to apply the power rule of integration.

Integrating the Simplified Expression

Alright, we're in the home stretch now! The integral has been simplified, and we're ready to integrate the expression -∫(u³ - u⁵)du. This is where we apply the power rule of integration. For each term, we add 1 to the power and divide by the new power. It's a straightforward process, but let's take it step by step to make sure we don't miss anything.

Applying the Power Rule of Integration

Integrating u³ gives us (u⁴)/4, and integrating u⁵ gives us (u⁶)/6. Don’t forget the negative sign outside the integral! We now have -(u⁴/4 - u⁶/6) + C. Where 'C' is the constant of integration. Remembering the constant of integration is vital; it signifies that there are infinitely many antiderivatives. Now, we distribute the negative sign: -(u⁴/4) + (u⁶/6) + C. So, the integral of u³ - u⁵ becomes (u⁶/6) - (u⁴/4) + C. We've successfully integrated the expression!

This is a critical step because it brings us close to the final answer. Each term gets integrated separately, and we systematically apply the power rule. The result is a cleaner, more manageable expression that allows us to find the antiderivative of the function. After this step, we are almost done!

Back-Substituting and Finding the Final Solution

We're almost there, guys! We've integrated the expression, but our answer is currently in terms of u. We need to substitute back cos(x) for u to get the final answer in terms of the original variable, x. This is the last and most important step to solve the integral.

Replacing u with cos(x)

Remember that u = cos(x). We can plug this back into our integrated expression: (u⁶/6) - (u⁴/4) + C. Replacing u with cos(x) gives us (cos⁶(x)/6) - (cos⁴(x)/4) + C. And there you have it! This is the indefinite integral of sin³(x)cos³(x). We've gone from a seemingly complicated expression to a clean, elegant solution, showing how the combination of trig identities, strategic substitutions, and integration techniques can simplify complex math problems.

We did it! We successfully calculated the integral of sin³(x)cos³(x). The answer is (cos⁶(x)/6) - (cos⁴(x)/4) + C. This solution isn't just a collection of symbols; it provides a relationship between the original function and its integral. This process demonstrates a powerful technique used in integral calculus.

Key Takeaways and Conclusion

Alright, let's sum up what we've learned and highlight some crucial points. This integral was a great example of how you can combine trigonometric identities with clever substitution to solve complex integrals. These techniques aren't just for this specific problem but are valuable tools for tackling other integrals.

Recap of the Steps

1. Rewrite: We began by rewriting the integral using a trigonometric identity. This helped to simplify the expression by transforming it into a more manageable form. We used the identity sin²(x) + cos²(x) = 1. This step transformed the integrand. Without this, the problem would be more difficult to solve. The rewrite step opens up possibilities for simplification.
2. Substitution: We employed substitution, letting u = cos(x), to transform the integral into a simpler form. This allowed us to work with a simpler expression, making integration more straightforward. The derivative of cos(x) was cleverly used to substitute and simplify the integral.
3. Integrate: We integrated the simplified expression using the power rule. This is the core of integration. We integrated term by term.
4. Back-Substitute: Finally, we substituted back cos(x) for u to get our final answer in terms of x. This gave us our final solution. This step is about going back to our original variable, which is crucial for answering the initial question. It's about getting back to our original problem's context.

Importance of Practice

This whole process might seem a bit daunting at first, but with practice, it becomes second nature. The more integrals you solve, the more comfortable you'll become with recognizing patterns and choosing the right techniques. So, keep practicing, and don't be afraid to try different approaches. Each integral is a new puzzle, and the satisfaction of solving it is well worth the effort. The value of practicing cannot be overstated. With practice, these steps become easier and the intuition for solving integrals strengthens.

Final Thoughts

So there you have it: the complete solution to the integral of sin³(x)cos³(x). We've explored the problem, broken it down into manageable steps, and used our mathematical tools to arrive at the final answer. Remember, the journey is just as important as the destination. The skills you've gained by working through this problem will be invaluable as you continue your calculus journey. Keep exploring, keep learning, and most importantly, keep having fun with math! You now have a solid understanding of how to solve this integral and, more importantly, a broader understanding of how to approach similar problems. Happy integrating, everyone!