Evaluating The Integral Of X²(x³-1)⁷ From 0 To 7

In the realm of calculus, definite integrals play a crucial role in determining the area under a curve between two specified limits. This article delves into the evaluation of a particular definite integral: ∫₀⁷ x²(x³-1)⁷ dx. We will explore the techniques and steps involved in solving this integral, providing a comprehensive understanding of the underlying concepts and methodologies. Mastering the evaluation of integrals like this is fundamental for students, engineers, and anyone working with mathematical models in various fields. Understanding the process not only reinforces calculus principles but also builds problem-solving skills applicable across diverse domains.

The integral ∫₀⁷ x²(x³-1)⁷ dx presents a classic example of a problem that can be efficiently solved using u-substitution, a powerful technique for simplifying integrals. The integrand, x²(x³-1)⁷, suggests a composite function where the derivative of the inner function (x³-1) is closely related to the x² term. This observation is key to recognizing the applicability of u-substitution. By choosing an appropriate substitution, we can transform the integral into a simpler form that is easier to evaluate. In the subsequent sections, we will meticulously walk through the steps, explaining each stage in detail to ensure clarity and comprehension.

Before diving into the solution, let's briefly touch upon the significance of definite integrals. Definite integrals are not just mathematical exercises; they have profound applications in physics, engineering, economics, and many other disciplines. For instance, they can be used to calculate displacement from velocity, work done by a force, consumer surplus in economics, and probabilities in statistics. The ability to accurately evaluate definite integrals is therefore an indispensable skill for anyone working in these areas. The given integral, while seemingly abstract, embodies the principles underlying these real-world applications. By understanding how to solve it, we gain a deeper appreciation for the broader utility of calculus in solving practical problems.

The given problem is to evaluate the definite integral ∫₀⁷ x²(x³-1)⁷ dx. This integral involves a polynomial function raised to a power, which suggests that a u-substitution might be an effective method for solving it. The limits of integration are from 0 to 7, indicating that we are looking for the definite value of the integral over this interval. The integrand consists of two parts: x² and (x³-1)⁷. The presence of x² as a factor and (x³-1) as a composite function is a strong indicator for considering u-substitution, as the derivative of x³-1 is proportional to x².

To approach this problem effectively, we need to recognize the structure of the integrand and identify a suitable substitution. The key observation is that the derivative of x³-1 is 3x², which is a constant multiple of the x² term present in the integrand. This relationship suggests that if we let u = x³-1, then du will involve x² dx, allowing us to simplify the integral. This method, known as u-substitution, is a fundamental technique in integral calculus used to simplify integrals involving composite functions. The goal is to transform the integral into a form that we can readily integrate using basic integration rules.

Once we perform the u-substitution, we will also need to change the limits of integration accordingly. Since the original limits are in terms of x, we need to convert them to the corresponding limits in terms of u. This is a crucial step in evaluating definite integrals using substitution, as it ensures that we are calculating the integral over the correct interval in the new variable. After evaluating the integral in terms of u, we can either substitute back for x and evaluate at the original limits, or we can directly evaluate the integral at the new limits in terms of u. Both methods will yield the same result, but using the new limits often simplifies the calculation.

Let's now proceed with solving the integral step by step. The first step is to choose a suitable substitution. As discussed earlier, the composite function (x³-1)⁷ suggests that letting u = x³-1 is a promising approach. This substitution simplifies the integrand and allows us to apply the power rule for integration more easily. Choosing the right substitution is crucial in u-substitution, and it often comes from recognizing composite functions and their derivatives within the integral.

Step 1: Perform u-Substitution:

Let u = x³-1. Then, the derivative of u with respect to x is du/dx = 3x². Solving for dx, we get dx = du / (3x²).

Now, substitute u and dx into the integral:

∫ x²(x³-1)⁷ dx = ∫ x²(u)⁷ (du / (3x²)).

Notice that the x² terms cancel out, simplifying the integral to:

(1/3) ∫ u⁷ du.

This simplified integral is now much easier to handle. The constant factor 1/3 can be pulled out of the integral, and we are left with the integral of a simple power function, u⁷.

Step 2: Change the Limits of Integration:

Since we are evaluating a definite integral, we need to change the limits of integration from x to u. The original limits are x = 0 and x = 7. We substitute these values into our substitution equation u = x³-1 to find the new limits.

When x = 0, u = (0)³-1 = -1. When x = 7, u = (7)³-1 = 343-1 = 342.

So, the new limits of integration are u = -1 and u = 342. This step is crucial for evaluating the definite integral correctly using the substituted variable.

Step 3: Evaluate the Integral in Terms of u:

Now we can rewrite the integral with the new limits and the simplified integrand:

(1/3) ∫₋₁³⁴² u⁷ du.

Using the power rule for integration, which states that ∫xⁿ dx = (x^(n+1)) / (n+1) + C, we integrate u⁷ with respect to u:

(1/3) [u⁸ / 8]₋₁³⁴².

Now we evaluate the antiderivative at the upper and lower limits:

(1/3) [(342)⁸ / 8 - (-1)⁸ / 8].

Step 4: Calculate the Final Value:

Calculate the values:

(1/3) [(342)⁸ / 8 - 1 / 8] = (1/24) [(342)⁸ - 1].

This gives us the final value of the definite integral. The expression (1/24) [(342)⁸ - 1] represents a large number, but it is the precise value of the integral.

While we have successfully evaluated the integral using the changed limits of integration, let's briefly discuss an alternative method: substituting back for x after finding the antiderivative in terms of u. This approach is equally valid and can be useful in cases where changing the limits might be cumbersome.

After finding the antiderivative in terms of u, we have:

(1/3) [u⁸ / 8] + C.

Substituting back u = x³-1, we get:

(1/3) [(x³-1)⁸ / 8] + C = [(x³-1)⁸ / 24] + C.

Now, we evaluate this expression at the original limits of integration, x = 0 and x = 7:

[((7)³-1)⁸ / 24] - [((0)³-1)⁸ / 24] = [(342)⁸ / 24] - [1 / 24] = (1/24) [(342)⁸ - 1].

As we can see, this method yields the same result as before. The choice between changing the limits and substituting back often depends on personal preference and the specific details of the integral.

In conclusion, we have successfully evaluated the definite integral ∫₀⁷ x²(x³-1)⁷ dx using the method of u-substitution. This technique is a powerful tool in integral calculus for simplifying integrals involving composite functions. By choosing an appropriate substitution (u = x³-1 in this case), we transformed the integral into a more manageable form and were able to apply the power rule for integration. We also discussed the importance of changing the limits of integration when evaluating definite integrals using substitution, as well as an alternative method of substituting back for the original variable.

The result, (1/24) [(342)⁸ - 1], demonstrates the power of calculus in providing precise solutions to complex problems. Definite integrals have wide-ranging applications in various fields, and mastering the techniques for evaluating them is essential for anyone working with mathematical models. The u-substitution method, as demonstrated in this article, is a fundamental skill that will be invaluable in tackling more advanced integration problems.

Understanding the nuances of integration, including when and how to apply specific techniques like u-substitution, is crucial for success in calculus and related disciplines. This article has provided a detailed walkthrough of the process, highlighting the key steps and considerations involved. By carefully following these steps and practicing with similar problems, you can build confidence and proficiency in evaluating definite integrals. The ability to solve such problems not only enhances your mathematical skills but also prepares you for applying calculus in real-world contexts, whether in engineering, physics, economics, or any other field that relies on mathematical modeling.