Solve ∫2x√(4x-5) Dx: U-Substitution Method

by Henrik Larsen 43 views

Hey there, math enthusiasts! Today, we're diving deep into the world of calculus to tackle a fascinating integral: ∫2x√(4x-5) dx. This integral might look a bit intimidating at first glance, but don't worry, we'll break it down step by step using a clever technique called u-substitution. So, buckle up and let's get started!

Understanding the Challenge: Why U-Substitution?

When we stare at ∫2x√(4x-5) dx, we notice a couple of things. First, we have a square root function, which can be a bit tricky to handle directly. Second, we have a product of x and √(4x-5), making it difficult to apply basic integration rules. This is where u-substitution comes to the rescue. U-substitution is a powerful method that allows us to simplify integrals by replacing a complex expression with a single variable, 'u'. The key is to choose our 'u' wisely. In this case, the expression inside the square root, 4x-5, seems like a promising candidate. Think of it like this, guys: we're trying to make the integral look simpler by substituting a chunk of it with a single variable. By making the right substitution, we can transform the integral into a form that we can easily integrate using standard techniques. It's like magic, but it's actually just good old calculus!

The beauty of u-substitution lies in its ability to untangle complex integrals. By strategically choosing a substitution, we can often transform an intimidating integral into a much simpler one that we can solve with ease. In this specific problem, the presence of the square root and the product of terms hints at the effectiveness of u-substitution. When you see a composite function (a function within a function), like the square root of (4x-5) here, it's a strong signal that u-substitution might be the way to go. The goal is to find a substitution that simplifies the integrand (the expression inside the integral) and makes the integration process more manageable. We're essentially trying to reverse the chain rule of differentiation, which is why identifying the "inner" function is so crucial. Once we've made our substitution, we'll need to adjust the differential (the dx part) accordingly, which we'll see in the next step. This process of transforming the integral is where the magic happens, turning a seemingly impossible problem into a solvable one.

The U-Substitution Process: A Step-by-Step Guide

Alright, let's get our hands dirty and actually perform the u-substitution. Here's how it works:

  1. Choose your u: As we discussed, let's set u = 4x - 5. This is the heart of the method, guys. We're picking a part of the integral that, when substituted, will simplify things significantly.

  2. Find du: Now, we need to find the derivative of u with respect to x, which we denote as du/dx. Differentiating u = 4x - 5, we get du/dx = 4. This step is crucial because it allows us to relate du and dx, which we'll need to do in the next step.

  3. Solve for dx: We need to express dx in terms of du. From du/dx = 4, we can write du = 4 dx. Dividing both sides by 4, we get dx = du/4. This is a key transformation that will allow us to replace dx in the original integral with an expression involving du. It's like translating from one language to another – we're changing the variable of integration from x to u.

  4. Rewrite the integral: Now, we substitute u and dx in the original integral: ∫2x√(4x-5) dx becomes ∫2x√u (du/4). But wait! We still have an x in the integral. We need to get rid of it. This is where we use our original substitution, u = 4x - 5, to solve for x. Adding 5 to both sides gives us u + 5 = 4x, and dividing by 4 gives us x = (u + 5)/4. Now we can substitute this into our integral:

    ∫2x√u (du/4) = ∫2((u + 5)/4)√u (du/4) = ∫(u + 5)/2 √u (du/4) = (1/8)∫(u + 5)√u du.

    See how we've successfully transformed the integral into one involving only u? That's the power of u-substitution in action!

Integrating with U: A Simpler Landscape

Now that we've made our substitution, the integral looks much friendlier. We have (1/8)∫(u + 5)√u du. To make it even easier, let's distribute the √u: (1/8)∫(u√u + 5√u) du. Remember that √u is the same as u^(1/2), so we can rewrite this as (1/8)∫(u^(3/2) + 5u^(1/2)) du. Now we're in familiar territory! We can 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.

Applying the power rule, we get:

(1/8)∫(u^(3/2) + 5u^(1/2)) du = (1/8) [(u^(5/2))/(5/2) + 5(u^(3/2))/(3/2)] + C

Simplifying, we have:

(1/8) [(2/5)u^(5/2) + (10/3)u^(3/2)] + C

Distributing the 1/8, we get:

(1/20)u^(5/2) + (5/12)u^(3/2) + C

We've successfully integrated with respect to u! But we're not done yet. Remember, we started with an integral in terms of x, so we need to convert back.

Returning to X: The Final Transformation

We've done the hard part – the integration itself. Now, we need to switch back from u to x. We know that u = 4x - 5, so we simply substitute this back into our expression:

(1/20)u^(5/2) + (5/12)u^(3/2) + C = (1/20)(4x - 5)^(5/2) + (5/12)(4x - 5)^(3/2) + C

And there you have it! That's the solution to the integral ∫2x√(4x-5) dx. We've successfully navigated the world of u-substitution and arrived at our final answer. But, as any good mathematician knows, we're not quite finished until we've considered simplifying our result.

Simplifying the Solution: Making It Shine

Our solution, (1/20)(4x - 5)^(5/2) + (5/12)(4x - 5)^(3/2) + C, is perfectly correct, but we can make it look even cleaner. Notice that both terms have a common factor of (4x - 5)^(3/2). Let's factor it out:

(1/20)(4x - 5)^(5/2) + (5/12)(4x - 5)^(3/2) + C = (4x - 5)^(3/2) [(1/20)(4x - 5) + (5/12)] + C

Now, let's simplify the expression inside the brackets. To add the fractions, we need a common denominator, which is 60:

(4x - 5)^(3/2) [(1/20)(4x - 5) + (5/12)] + C = (4x - 5)^(3/2) [(3(4x - 5) + 25)/60] + C

Distributing the 3 in the numerator, we get:

(4x - 5)^(3/2) [(12x - 15 + 25)/60] + C = (4x - 5)^(3/2) [(12x + 10)/60] + C

Finally, we can simplify the fraction by dividing both the numerator and denominator by 2:

(4x - 5)^(3/2) [(12x + 10)/60] + C = (4x - 5)^(3/2) [(6x + 5)/30] + C

So, our simplified solution is:

[(6x + 5)/30] (4x - 5)^(3/2) + C

This is the same answer as before, just presented in a more elegant and concise form. Simplifying your solution is always a good practice, as it can make it easier to work with in future calculations and helps you spot any potential errors. Plus, it just looks nicer, right?

Conclusion: Mastering the Art of U-Substitution

Guys, we've successfully conquered the integral ∫2x√(4x-5) dx using the powerful technique of u-substitution! We saw how choosing the right 'u' can transform a seemingly complex integral into a much simpler one. We also learned the importance of converting back to the original variable and simplifying our solution. U-substitution is a fundamental tool in calculus, and mastering it will open doors to solving a wide range of integrals. So, keep practicing, and you'll become a pro in no time! Remember, the key is to identify the "inner" function and make the appropriate substitution. With a little practice, you'll be able to spot these opportunities and tackle even the most challenging integrals with confidence. Now go forth and integrate!