Solving ∫₁^√³ X 4^(x²) Dx A Step-by-Step Guide

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Introduction

Hey guys! Today, we're diving into the fascinating world of calculus to tackle a definite integral. Specifically, we're going to evaluate the integral ∫₁^√³ x 4^(x²) dx. Integrals might seem intimidating at first, but with the right techniques, they become much more manageable. This integral is a perfect example of how u-substitution, a powerful method, can simplify complex expressions and lead us to a solution. So, grab your pencils and let's get started!

Definite integrals are a cornerstone of calculus, representing the accumulated effect of a function over a specific interval. They have countless applications in physics, engineering, economics, and many other fields. Whether it's calculating the area under a curve, the work done by a force, or the probability of an event, definite integrals provide the tools we need to solve real-world problems. Before we jump into the solution, let's take a moment to understand the problem at hand. We have the integral of x multiplied by 4 raised to the power of x², all evaluated from 1 to √3. This looks a bit tricky at first glance, but don't worry! We'll break it down step by step.

Understanding the Integral

To truly grasp the integral, it helps to visualize what it represents. The function x * 4^(x²) is a curve, and the integral ∫₁^√³ x 4^(x²) dx represents the area under this curve between the limits x = 1 and x = √3. This area is a definite quantity, a specific number, which we aim to find. Notice the complexity of the function. We have a variable x both as a multiplier and within the exponent. This is where u-substitution comes to our rescue. By carefully choosing a substitution, we can transform this integral into a simpler form that we can easily solve. The key to successful integration often lies in recognizing patterns and selecting the appropriate technique. In this case, the presence of x² in the exponent and x as a separate term hints at the possibility of using u-substitution. Think of u-substitution as a way to "undo" the chain rule of differentiation. It allows us to simplify expressions by replacing a complex function with a single variable, making the integral more approachable.

The Power of U-Substitution

Alright, let's talk strategy. The key to solving this integral lies in a clever technique called u-substitution. The goal of u-substitution is to simplify the integral by replacing a complex part of the function with a new variable, u. This often transforms the integral into a more recognizable form that we can integrate directly. In our case, we notice that the exponent of 4 is x², and the derivative of x² is 2x, which is similar to the x term in the integral. This is a strong indication that u-substitution will work wonders here. We'll let u = x², and this substitution will simplify the exponential term significantly. Remember, the beauty of u-substitution is that it allows us to deal with the inner function separately, making the integration process smoother. The trick is to identify the “inner” function and its derivative within the integral. Once we have our u, we need to find du, which is the derivative of u with respect to x, multiplied by dx. This step is crucial because it allows us to replace the original differential dx with an expression involving du. By transforming both the function and the differential, we ensure that our new integral is entirely in terms of u, making it solvable.

Step-by-Step Solution

Let's break down the solution step-by-step:

  1. Choose u: Let u = x². This is our pivotal substitution. We're essentially replacing the complicated exponent with a single variable.
  2. Find du: Differentiate u with respect to x: du/dx = 2x. Then, solve for dx: dx = du / (2x). Now we have our differential transformation.
  3. Substitute: Replace x² with u and dx with du / (2x) in the integral: ∫ x * 4^u * (du / (2x)). Notice how the x terms cancel out, leaving us with a much simpler integral.
  4. Simplify: After canceling the x terms and pulling out the constant 1/2, our integral becomes: (1/2) ∫ 4^u du. This is now a standard exponential integral.
  5. Integrate: The integral of 4^u with respect to u is 4^u / ln(4). So, we have (1/2) * (4^u / ln(4)) + C. Don't forget the constant of integration C for indefinite integrals, but remember, we are dealing with a definite integral here, so we'll handle the limits of integration shortly.
  6. Substitute back: Replace u with x²: (1/2) * (4^(x²) / ln(4)). We've now transformed our solution back into terms of x.
  7. Evaluate the definite integral: Now, we need to evaluate this expression at the limits of integration, x = √3 and x = 1. This means plugging in these values for x and subtracting the results: [(1/2) * (4^(√3)²) / ln(4)] - [(1/2) * (4^(1²) / ln(4)].
  8. Simplify: Simplify the expression: [(1/2) * (4³) / ln(4)] - [(1/2) * (4) / ln(4)] = (1/2) * (64 / ln(4)) - (1/2) * (4 / ln(4)) = (1/2) * (60 / ln(4)) = 30 / ln(4).
  9. Final Answer: The value of the integral ∫₁^√³ x 4^(x²) dx is 30 / ln(4). We've successfully solved the integral using u-substitution!

Detailed Breakdown of Steps

Let's dive deeper into each step to ensure we understand the logic and mechanics behind the solution. The heart of this problem lies in the u-substitution technique, so let's dissect it further.

1. Choosing 'u'

The success of u-substitution hinges on selecting the right 'u'. In our case, u = x² was the perfect choice because its derivative, 2x, appears (or at least a multiple of it appears) in the integral. This is the key indicator that u-substitution will simplify the expression. If we had chosen a different 'u', such as u = 4^(x²), the substitution wouldn't have led to a simpler integral. The golden rule is to look for an “inner” function whose derivative is also present in the integral. Think of it as reversing the chain rule – we're trying to undo a differentiation that involved the chain rule.

2. Finding 'du'

Once we have 'u', we need to find 'du'. This involves differentiating 'u' with respect to 'x' and then solving for 'dx'. In our case, du/dx = 2x, so dx = du / (2x). This step is crucial because it allows us to replace the original differential 'dx' with an expression involving 'du'. This transformation ensures that our new integral is entirely in terms of 'u'. Without this step, we wouldn't be able to integrate with respect to 'u'.

3. Substitution

Now comes the substitution magic! We replace every instance of x² with 'u' and 'dx' with 'du / (2x)' in the original integral. This step transforms the integral from ∫ x * 4^(x²) dx to ∫ x * 4^u * (du / (2x)). Notice the beautiful cancellation that occurs – the 'x' terms cancel out, leaving us with a much simpler integral. This cancellation is a hallmark of successful u-substitution. It indicates that we've chosen the right 'u' and that the substitution is working as intended.

4. Simplifying the Integral

After the cancellation, we're left with ∫ 4^u * (1/2) du. We can pull the constant 1/2 out of the integral, resulting in (1/2) ∫ 4^u du. This integral is now in a standard form that we can easily integrate. The simplification highlights the power of u-substitution – it transforms a complex integral into a manageable one.

5. Integration

The integral of 4^u with respect to 'u' is 4^u / ln(4). This is a standard exponential integral, and knowing these basic integral forms is essential for solving calculus problems. So, we have (1/2) * (4^u / ln(4)) + C. Remember the '+ C' for indefinite integrals, representing the constant of integration. However, since we're dealing with a definite integral, we'll handle the limits of integration in a later step.

6. Substituting Back

We're not done yet! We need to substitute back x² for 'u' to express our solution in terms of the original variable, 'x'. This gives us (1/2) * (4^(x²) / ln(4)). This step is crucial because it returns our solution to the original context of the problem. We started with an integral in terms of 'x', and our final answer should also be in terms of 'x'.

7. Evaluating the Definite Integral

Now comes the final step – evaluating the definite integral at the limits of integration, x = √3 and x = 1. This involves plugging in these values for 'x' into our expression and subtracting the results: [(1/2) * (4^(√3)²) / ln(4)] - [(1/2) * (4^(1²) / ln(4)]. This step gives us a numerical value for the definite integral, representing the area under the curve between the specified limits.

8. Simplification and Final Answer

Finally, we simplify the expression: [(1/2) * (4³) / ln(4)] - [(1/2) * (4) / ln(4)] = (1/2) * (64 / ln(4)) - (1/2) * (4 / ln(4)) = (1/2) * (60 / ln(4)) = 30 / ln(4). So, the value of the integral ∫₁^√³ x 4^(x²) dx is 30 / ln(4). We've successfully navigated the integral using u-substitution and arrived at our final answer!

Conclusion

So there you have it! We've successfully evaluated the integral ∫₁^√³ x 4^(x²) dx using the u-substitution technique. This powerful method allowed us to simplify a complex integral into a manageable form, leading us to the solution 30 / ln(4). Remember, guys, practice makes perfect! The more integrals you solve, the better you'll become at recognizing patterns and applying the appropriate techniques. Keep exploring the world of calculus, and you'll discover its endless possibilities. Whether it's calculating areas, volumes, or solving differential equations, the principles of integration are fundamental to many areas of science and engineering. So, embrace the challenge, keep practicing, and you'll become an integration master in no time! We encourage you to try similar problems and experiment with different substitutions. The key is to understand the underlying concepts and apply them creatively. Happy integrating!