Solve ∫ 5 Dx / √(3 - 3x^2): A Step-by-Step Guide
Hey guys! Today, we're going to dive into a super interesting integral problem. Integrals can seem daunting at first, but trust me, breaking them down step-by-step makes them much more manageable. We're tackling the integral ∫ 5 dx / √(3 - 3x^2). Sounds fun, right? Let’s get started and make sure we understand every little detail along the way. Remember, practice makes perfect, so by the end of this guide, you'll not only know how to solve this specific integral but also feel more confident tackling similar problems. Let’s make math less scary and more… well, enjoyable! This problem involves several key techniques that are fundamental to calculus, such as trigonometric substitution and constant manipulation. By mastering these techniques, you'll be well-equipped to handle a wide range of integrals. So, grab your favorite beverage, find a comfy spot, and let’s jump into the world of calculus together! We'll explore each step with clear explanations and insights, ensuring that you grasp the underlying concepts rather than just memorizing formulas. Throughout this guide, we'll emphasize the importance of recognizing patterns and choosing the appropriate methods. This skill is crucial not only for solving integrals but also for tackling various mathematical problems. Are you ready to unravel the mysteries of this integral and boost your calculus prowess? Let’s dive in and conquer this challenge together!
1. Initial Setup and Simplification
Okay, so the integral we're dealing with is ∫ 5 dx / √(3 - 3x^2). The first thing we want to do is simplify this bad boy as much as possible. Notice that we have a constant, 5, in the numerator. We can pull that right out of the integral. This is a basic property of integrals: ∫ c * f(x) dx = c ∫ f(x) dx, where c is a constant. Pulling out constants makes the integral cleaner and easier to work with. Think of it like decluttering your workspace before starting a big project – it just makes things smoother. So, our integral now looks like this: 5 ∫ dx / √(3 - 3x^2). Much better, right? Next, let’s focus on the denominator, √(3 - 3x^2). We can factor out a 3 from inside the square root. Factoring is a super useful trick in calculus, especially when dealing with square roots or polynomials. Factoring out the 3 gives us √(3(1 - x^2)). Now, remember the property of square roots: √(a * b) = √a * √b. So, we can rewrite √(3(1 - x^2)) as √3 * √(1 - x^2). This is a key step because it isolates the x^2 term and makes the integral look more familiar. Our integral now looks like: 5 ∫ dx / (√3 * √(1 - x^2)). We still have a constant, √3, in the denominator. Just like we pulled out the 5, we can pull out the √3 as well, but this time it will be in the denominator. So we get: (5 / √3) ∫ dx / √(1 - x^2). Now, this is starting to look like something we recognize! This simplified form is much easier to handle and sets us up perfectly for the next step, which involves trigonometric substitution. These initial simplification steps are super important because they transform a complex-looking integral into a more manageable form. Always look for ways to simplify before jumping into more advanced techniques. It can save you a lot of headaches down the road.
2. Trigonometric Substitution
Alright, guys, here’s where the magic happens! Our integral is now (5 / √3) ∫ dx / √(1 - x^2). This form screams trigonometric substitution. Specifically, it looks a lot like the derivative of arcsin(x), which involves a √(1 - x^2) in the denominator. When you see something like √(a^2 - x^2), where a is a constant, the classic trig substitution is x = a * sin(θ). In our case, a = 1, so we're going to use x = sin(θ). This substitution might seem a bit mysterious at first, but trust me, it’s a powerful technique that simplifies many integrals involving square roots. By substituting x with sin(θ), we're essentially transforming our integral into a trigonometric one, which is often easier to solve. Now, if x = sin(θ), we need to find dx in terms of dθ. To do this, we take the derivative of both sides with respect to θ. The derivative of sin(θ) is cos(θ), so we get dx = cos(θ) dθ. This is a crucial part of the substitution process. We’re not just changing x; we’re also changing dx to keep the integral consistent. Okay, let’s substitute x = sin(θ) and dx = cos(θ) dθ into our integral. We get: (5 / √3) ∫ cos(θ) dθ / √(1 - sin^2(θ)). Now, we need to simplify the denominator. Remember the trigonometric identity sin^2(θ) + cos^2(θ) = 1? We can rearrange this to get cos^2(θ) = 1 - sin^2(θ). So, √(1 - sin^2(θ)) becomes √(cos^2(θ)), which simplifies to |cos(θ)|. For the sake of simplicity, we'll assume cos(θ) is positive in our interval, so we can just write cos(θ). Our integral now looks like: (5 / √3) ∫ cos(θ) dθ / cos(θ). Look at that! The cos(θ) terms cancel out, leaving us with (5 / √3) ∫ dθ. This is a much simpler integral to deal with. Trigonometric substitution might seem like a complicated trick, but it’s all about recognizing patterns and using the right identities to simplify the integral. With practice, you’ll get really good at spotting these patterns and choosing the appropriate substitutions. This step is a perfect example of how a clever substitution can turn a seemingly difficult integral into a straightforward one. The cancellation of the cosine terms is a beautiful result of this technique.
3. Integrating with Respect to θ
Alright, awesome! We've simplified the integral down to (5 / √3) ∫ dθ. Now, this is an integral we can definitely handle! Integrating dθ with respect to θ is super straightforward. Remember, the integral of a constant (in this case, 1) is just the variable of integration (θ) plus a constant of integration (C). So, ∫ dθ = θ + C. Therefore, our integral (5 / √3) ∫ dθ becomes (5 / √3) * (θ + C). We can distribute the (5 / √3) to get (5 / √3)θ + (5 / √3)C. But remember, (5 / √3)C is just another constant, so we can simply write it as C. Our result so far is (5 / √3)θ + C. Now, we’re not quite done yet. We need to get back to our original variable, x. Remember our trigonometric substitution? We said x = sin(θ). We need to solve for θ in terms of x. To do this, we take the inverse sine (arcsin) of both sides: θ = arcsin(x). This is a crucial step. We’ve successfully integrated with respect to θ, but the problem was originally in terms of x, so we need to convert back. Substituting θ = arcsin(x) into our result, we get (5 / √3)arcsin(x) + C. And there you have it! We've solved the integral. This step is a great reminder that integrating in terms of a new variable is only half the battle. The real key is to always remember to convert back to the original variable. This ensures that your final answer is expressed in the same terms as the original problem. The constant of integration, C, is also a critical part of the solution. Always remember to include it, as it represents the family of functions that have the same derivative.
4. Final Answer and Simplification
Okay, guys, we're almost at the finish line! Our solution right now is (5 / √3)arcsin(x) + C. This is technically correct, but we can make it look even nicer by rationalizing the denominator. Rationalizing the denominator means getting rid of any square roots in the denominator. In our case, we have √3 in the denominator. To rationalize it, we multiply both the numerator and the denominator by √3. So, (5 / √3) becomes (5√3) / (√3 * √3) = (5√3) / 3. Now our solution looks like (5√3 / 3)arcsin(x) + C. This is the simplified, final answer. It’s cleaner, more elegant, and generally preferred in mathematical notation. This final simplification step is often overlooked, but it’s important for presenting your answer in the best possible form. Rationalizing the denominator is a common practice in mathematics, and it shows attention to detail. So, let's recap what we did. We started with ∫ 5 dx / √(3 - 3x^2). We simplified by pulling out constants and factoring. Then we used trigonometric substitution (x = sin(θ)) to transform the integral. We integrated with respect to θ and converted back to x using arcsin(x). Finally, we rationalized the denominator to get our final answer: (5√3 / 3)arcsin(x) + C. Awesome job, guys! You've tackled a pretty complex integral, and hopefully, you feel a lot more confident about these types of problems now. Remember, practice is key. The more integrals you solve, the better you'll get at recognizing patterns and applying the right techniques. So keep practicing, and you'll become an integral-solving pro in no time! This problem highlights the power of strategic simplification and substitution. By breaking down the integral into manageable steps and using the right techniques, we were able to arrive at a clear and concise solution. Keep these strategies in mind as you tackle future calculus challenges.
5. Key Takeaways and Practice Problems
So, what are the main takeaways from solving this integral? First, always look for opportunities to simplify the integral. Pull out constants, factor out terms, and see if you can rewrite the integral in a more manageable form. Simplification can make a huge difference in the complexity of the problem. Next, recognize patterns that suggest specific techniques. In this case, the √(a^2 - x^2) form strongly suggests trigonometric substitution using x = a * sin(θ). Knowing these patterns can save you a lot of time and effort. Then, trigonometric substitution is a powerful tool, but remember to convert back to the original variable after integrating. Don't forget that crucial step! And finally, always simplify your final answer as much as possible. Rationalize denominators and combine like terms to present your solution in its best form. These steps are essential for clear and accurate communication of your results. To help you practice, here are a few similar problems you can try:
- ∫ 3 dx / √(4 - 4x^2)
- ∫ 2 dx / √(9 - 9x^2)
- ∫ 7 dx / √(1 - x^2)
Try solving these integrals using the same techniques we covered in this guide. Remember, the more you practice, the better you'll become at recognizing patterns and applying the appropriate methods. Solving integrals is like learning a new language – the more you use it, the more fluent you become. Don't be afraid to make mistakes; they're part of the learning process. And if you get stuck, go back and review the steps we took in this guide. Each of these practice problems is designed to reinforce the concepts and techniques we’ve discussed. By working through them, you’ll solidify your understanding and build your confidence. Remember, calculus is a skill that improves with practice, so keep challenging yourself and exploring new problems. With dedication and perseverance, you’ll master the art of integration and unlock a whole new world of mathematical possibilities. So, grab your pencil, paper, and let’s keep solving!
