Solve ∫(x+1)√(2x+x²) Dx From 0 To 1: A Step-by-Step Guide
Hey guys! Today, we're diving into a super interesting calculus problem: solving the definite integral of (x+1)√(2x+x²) from 0 to 1. This problem might seem a bit intimidating at first, but trust me, we'll break it down step by step and make it crystal clear. So, grab your pencils, and let's get started!
Understanding the Integral
Before we jump into the solution, let's quickly recap what a definite integral actually represents. A definite integral essentially calculates the area under a curve between two specified limits. In our case, we want to find the area under the curve of the function f(x) = (x+1)√(2x+x²) between the points x = 0 and x = 1. This means we're looking for the accumulated value of this function over the interval [0, 1]. Integrals are fundamental in calculus and have wide applications in physics, engineering, economics, and many other fields. They allow us to calculate quantities like displacement, work, and probability, which makes understanding how to solve them crucial for anyone studying these disciplines. The beauty of calculus lies in its ability to transform complex problems into manageable steps, and this integral is a perfect example of that.
Why This Integral is Interesting
This particular integral is interesting because it's not immediately obvious how to solve it using basic integration techniques. We can't just directly apply the power rule or a simple trigonometric substitution. Instead, we need to employ a clever strategy – u-substitution – to simplify the integrand and make it solvable. Problems like these are fantastic for honing our problem-solving skills and deepening our understanding of calculus. They challenge us to think outside the box and recognize patterns that might not be immediately apparent. Plus, mastering techniques like u-substitution is essential for tackling more advanced integrals down the road. So, by working through this example, we're not just solving one problem; we're building a solid foundation for future calculus endeavors. The complexity of the integrand, (x+1)√(2x+x²), necessitates a strategic approach, which makes it an excellent exercise in calculus problem-solving.
The Strategy: U-Substitution
The key to cracking this integral is recognizing that the expression inside the square root, 2x + x², and the term (x+1) are related. If we differentiate 2x + x², we get 2 + 2x, which is just 2(x+1). This relationship suggests that u-substitution is the perfect technique for this problem. U-substitution is a powerful method that allows us to simplify integrals by replacing a complex expression with a single variable, 'u'. It's like a mathematical magic trick that transforms a daunting integral into something much more manageable. The idea behind u-substitution is to identify a function and its derivative within the integral, which then allows us to change the variable of integration and simplify the problem. By carefully choosing our 'u', we can often unravel complex integrands and reduce them to standard forms that we know how to integrate.
Defining 'u'
Let's define u = 2x + x². This is a crucial step because the right choice of 'u' can make or break the solution. In this case, we're choosing the expression inside the square root because its derivative is closely related to the term outside the square root. This is a common strategy in u-substitution: look for expressions whose derivatives also appear in the integrand. Now, let's find the derivative of u with respect to x: du/dx = 2 + 2x. We can rewrite this as du = (2x + 2) dx, or even better, du = 2(x + 1) dx. This is great news because we have (x+1) dx in our original integral! We can isolate (x+1) dx by dividing both sides by 2, giving us (1/2) du = (x + 1) dx. This substitution will significantly simplify our integral, making it much easier to solve. By carefully selecting u = 2x + x², we've set the stage for a smooth integration process, leveraging the relationship between the expression and its derivative to simplify the integral.
Transforming the Integral
Now that we've defined our 'u' and found the relationship between du and dx, let's rewrite the integral in terms of 'u'. Remember, our original integral was ∫(from 0 to 1) (x+1)√(2x+x²) dx. We're going to replace 2x + x² with 'u' and (x+1) dx with (1/2) du. This transformation will change the integral from being in terms of 'x' to being in terms of 'u', which is a crucial step in the u-substitution process. By doing this, we're essentially changing the coordinate system of our integration, making the problem simpler to visualize and solve. The goal is to create an integral that is easier to handle, often involving standard forms that we readily recognize.
Changing the Limits of Integration
But wait, there's a little detail we need to address! Since we're changing the variable of integration, we also need to change the limits of integration. Our original limits were x = 0 and x = 1. We need to find the corresponding values of 'u' for these x-values. When x = 0, u = 2(0) + (0)² = 0. So, our new lower limit is u = 0. When x = 1, u = 2(1) + (1)² = 3. So, our new upper limit is u = 3. This is a crucial step in definite integrals because we're calculating the area under the curve between specific points. Changing the limits ensures that we're calculating the area over the correct interval in the new variable. Forgetting to change the limits is a common mistake, so always remember to do this when using u-substitution with definite integrals. By correctly transforming the limits, we maintain the integrity of the integral and ensure an accurate result.
The New Integral
With these substitutions, our integral becomes ∫(from 0 to 3) (1/2)√u du. Notice how much simpler this looks compared to our original integral! We've successfully transformed a complex integral into a much more manageable form. The (1/2) factor is a constant, and we can pull it out of the integral, giving us (1/2) ∫(from 0 to 3) √u du. Now, we're dealing with the integral of √u, which is a standard form that we can easily handle using the power rule for integration. This transformation highlights the power of u-substitution: it allows us to break down complex problems into simpler, more solvable components. By rewriting the integral in terms of 'u', we've cleared the path for a straightforward integration process, making the solution much more accessible.
Solving the Simplified Integral
Now, let's tackle the simplified integral: (1/2) ∫(from 0 to 3) √u du. We know that √u is the same as u^(1/2), so we can rewrite the integral as (1/2) ∫(from 0 to 3) u^(1/2) du. To integrate u^(1/2), we use the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. In our case, n = 1/2, so n + 1 = 3/2. Applying the power rule, we get ∫u^(1/2) du = (u^(3/2))/(3/2) = (2/3)u^(3/2). This is a fundamental step in calculus, and mastering the power rule is essential for solving a wide range of integrals. By correctly applying the power rule, we've found the antiderivative of √u, which is a crucial step towards finding the definite integral.
Evaluating the Antiderivative
So, our integral becomes (1/2) * (2/3)u^(3/2) evaluated from 0 to 3. We can simplify this to (1/3)u^(3/2) evaluated from 0 to 3. Now, we need to plug in our limits of integration. First, we'll plug in the upper limit, u = 3, and then subtract the result of plugging in the lower limit, u = 0. This process is the fundamental theorem of calculus in action: we evaluate the antiderivative at the upper and lower limits and subtract the results to find the definite integral. It's a beautiful and powerful concept that connects differentiation and integration, allowing us to calculate areas and other accumulated quantities with ease. The careful evaluation of the antiderivative at the limits is crucial for obtaining the correct numerical answer.
Final Calculation
Plugging in u = 3, we get (1/3)(3)^(3/2) = (1/3)(3√3) = √3. Plugging in u = 0, we get (1/3)(0)^(3/2) = 0. So, our final answer is √3 - 0 = √3. There you have it! The definite integral of (x+1)√(2x+x²) from 0 to 1 is √3. This result represents the exact area under the curve of the function f(x) = (x+1)√(2x+x²) between x = 0 and x = 1. It's a concrete value that we've obtained through a series of careful steps, demonstrating the power and precision of calculus. By successfully solving this integral, we've not only found the area under the curve but also reinforced our understanding of u-substitution and the fundamental theorem of calculus.
Conclusion
Guys, we did it! We successfully solved the definite integral of (x+1)√(2x+x²) from 0 to 1. We saw how u-substitution can simplify complex integrals and how important it is to change the limits of integration when dealing with definite integrals. Remember, practice makes perfect, so keep tackling those integrals! Understanding and mastering these techniques is essential for anyone diving deeper into calculus and its applications. Each integral solved is a step forward in building a strong foundation in mathematical problem-solving. Keep exploring, keep practicing, and you'll become a calculus whiz in no time! And that's a wrap for this problem – awesome work, everyone!
