Solving ∫5x(x²+2)³ Dx: A Step-by-Step Guide

Dec 3, 2025 by Alex Johnson 44 views

Introduction to Integration

In the realm of calculus, integration is a fundamental operation with wide-ranging applications across various scientific and engineering disciplines. At its core, integration is the reverse process of differentiation, allowing us to find the area under a curve, determine volumes, and solve differential equations. This article delves into the step-by-step process of solving a specific integral: ∫5x(x²+2)³ dx. Whether you're a student grappling with calculus concepts or a professional seeking a refresher, this guide will provide a comprehensive understanding of the techniques involved.

To truly grasp the power of integration, it's essential to first understand its relationship with differentiation. Differentiation helps us find the rate of change of a function, while integration helps us find the function when we know its rate of change. Imagine you have a car traveling at varying speeds; differentiation can tell you the instantaneous acceleration at any point in time, while integration can tell you the total distance traveled over a certain period. This inverse relationship is the cornerstone of the fundamental theorem of calculus, a cornerstone of mathematical analysis.

Integration isn't just a theoretical concept; it's a practical tool used in a multitude of real-world applications. In physics, it helps calculate displacement from velocity, work done by a force, and the center of mass of an object. In economics, it's used to determine consumer surplus and producer surplus. In statistics, it plays a crucial role in probability calculations. The ability to compute integrals accurately and efficiently is thus an invaluable skill for anyone working in these fields. As we proceed through this guide, keep in mind the broader context of integration and its relevance to solving real-world problems. The specific integral we'll tackle, ∫5x(x²+2)³ dx, provides a perfect example of how substitution techniques can simplify complex expressions and lead to elegant solutions. So, let's embark on this journey and unravel the intricacies of this integral together.

Understanding the Integral: ∫5x(x²+2)³ dx

Before diving into the solution, let's break down the integral ∫5x(x²+2)³ dx to understand its components and why a specific technique like u-substitution is particularly effective here. The integral consists of several key parts: the integral symbol (∫), the integrand (5x(x²+2)³), the variable of integration (dx), and the limits of integration (which are not explicitly defined here, indicating we're looking for the indefinite integral, i.e., the general antiderivative). The integrand, 5x(x²+2)³, is the function we're trying to integrate. It's a product of two terms: 5x and (x²+2)³, making it a composite function. Recognizing this composition is crucial for choosing the right integration strategy.

The power of substitution shines when dealing with composite functions. Here, the expression (x²+2) is raised to the power of 3, making it a prime candidate for substitution. The derivative of x²+2 is 2x, which is present (or nearly present) in the other term, 5x, of the integrand. This relationship between the inner function (x²+2) and its derivative suggests that u-substitution will significantly simplify the integral. Substitution allows us to replace a complex expression with a single variable, making the integral more manageable. It's like having a tangled knot; instead of trying to untangle the whole thing at once, you identify a key loop and pull it out, simplifying the overall structure.

Why not just expand (x²+2)³ and then integrate term by term? While that's a valid approach, it would lead to a more cumbersome process. Expanding the expression would result in a polynomial with several terms, each needing to be integrated separately. This method increases the chances of making algebraic errors and generally takes more time. The beauty of u-substitution lies in its efficiency and elegance. It transforms the integral into a simpler form that can be readily solved using basic integration rules. By recognizing the structure of the integrand and strategically applying u-substitution, we can bypass the tedious expansion and arrive at the solution much faster. In the following sections, we'll walk through the steps of u-substitution in detail, demonstrating how this technique transforms a seemingly complex integral into a straightforward problem.

Step 1: Identify the 'u' and Compute 'du'

The first critical step in u-substitution is identifying the appropriate expression to designate as 'u'. This choice can significantly impact the simplicity of the resulting integral. In our case, the integral is ∫5x(x²+2)³ dx. As discussed earlier, the presence of a composite function, (x²+2)³, suggests that the inner function, x²+2, is a good candidate for 'u'. This is because its derivative, 2x, is closely related to the 5x term in the integrand. Choosing u = x²+2 allows us to simplify the expression significantly.

Now that we've identified 'u', the next step is to compute 'du', which represents the differential of 'u'. The differential is found by taking the derivative of 'u' with respect to x and then multiplying by dx. Mathematically, if u = f(x), then du = f'(x) dx. Applying this to our chosen 'u', we have:

u = x² + 2
du/dx = 2x
du = 2x dx

This relationship between 'du' and 'dx' is crucial for the substitution process. We need to be able to replace parts of the original integral with 'u' and 'du'. Notice that our integral contains 5x dx, while 'du' is 2x dx. This means we'll need to manipulate 'du' slightly to match the expression in the integral. This is a common occurrence in u-substitution and is easily handled by multiplying or dividing 'du' by a constant.

The ability to correctly identify 'u' and compute 'du' is the cornerstone of successful u-substitution. A wrong choice of 'u' can lead to a more complicated integral, or even an unsolvable one. Therefore, it's essential to carefully examine the integrand and look for composite functions and relationships between functions and their derivatives. With practice, you'll develop an intuition for choosing the right 'u' and setting up the substitution effectively. In the next step, we'll see how to use this 'du' to rewrite our integral in terms of 'u', bringing us closer to the final solution.

Step 2: Rewrite the Integral in Terms of 'u'

With 'u' and 'du' determined, we can now rewrite the original integral ∫5x(x²+2)³ dx in terms of 'u'. This step is where the magic of u-substitution truly happens, transforming a complex integral into a simpler one. Recall that we have:

u = x² + 2
du = 2x dx

Our integral contains 5x dx, but 'du' is 2x dx. To make the substitution work, we need to express 5x dx in terms of 'du'. We can do this by solving for x dx in the 'du' equation and then multiplying by the necessary constant. From du = 2x dx, we get:

x dx = du/2

Now we have an expression for x dx. However, our integral has 5x dx, so we need to multiply both sides of the equation by 5/2:

(5/2) du = 5x dx

Now we have the exact expression we need to substitute into the integral. We can replace 5x dx with (5/2) du and (x²+2) with u. This gives us the transformed integral:

∫5x(x²+2)³ dx = ∫u³ (5/2) du

Notice how the integral has become significantly simpler. We've eliminated the composite function and are left with a power of 'u' multiplied by a constant. This integral is much easier to evaluate using the power rule for integration. The constant (5/2) can be pulled out of the integral, further simplifying the expression:

∫u³ (5/2) du = (5/2) ∫u³ du

This transformed integral is now in a form that we can readily integrate. The key to this simplification was recognizing the relationship between the inner function and its derivative and strategically manipulating 'du' to match the expression in the original integral. By rewriting the integral in terms of 'u', we've effectively bypassed the complexity of the original integrand. In the next step, we'll apply the power rule to evaluate this simplified integral and find the antiderivative in terms of 'u'.

Step 3: Integrate with Respect to 'u'

After successfully rewriting the integral in terms of 'u', we are now ready to perform the integration. Our transformed integral is (5/2) ∫u³ du. This integral is a straightforward application of the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1 and C is the constant of integration. Applying the power rule to our integral, we have:

(5/2) ∫u³ du = (5/2) * (u³⁺¹)/(3+1) + C

Simplifying the expression, we get:

(5/2) * (u⁴/4) + C = (5/8)u⁴ + C

This is the antiderivative in terms of 'u'. The constant of integration, C, is crucial because the derivative of a constant is always zero. This means that there are infinitely many functions that have the same derivative, differing only by a constant. We include C to represent all possible antiderivatives. Integrating with respect to 'u' was a much simpler task than integrating the original integrand directly. The u-substitution technique allowed us to transform a complex integral into a basic power rule integration. This highlights the power and efficiency of substitution in calculus. However, we're not quite finished yet. Our original problem was in terms of 'x', so we need to convert our answer back to 'x'.

In the next step, we'll substitute back the original expression for 'u' in terms of 'x', giving us the final antiderivative in terms of the original variable. This step is essential for providing a complete solution to the integral. It's a reminder that u-substitution is a technique to simplify the integration process, but the final answer must be expressed in the context of the original problem.

Step 4: Substitute Back to 'x'

We've successfully integrated with respect to 'u' and found the antiderivative (5/8)u⁴ + C. However, the original integral was in terms of 'x', so we need to substitute back to express our answer in terms of 'x'. This final step ensures that our solution is in the same context as the original problem.

Recall that we defined u = x² + 2. To substitute back, we simply replace 'u' in our antiderivative with this expression:

(5/8)u⁴ + C = (5/8)(x² + 2)⁴ + C

This is the final solution to the integral ∫5x(x²+2)³ dx. We have found the general antiderivative, which represents a family of functions that differ only by a constant. The (5/8)(x² + 2)⁴ term represents the specific function we found through integration, and the '+ C' accounts for the constant of integration.

This step of substituting back highlights the importance of keeping track of the original variables throughout the u-substitution process. While the intermediate steps involve 'u', the final answer must be expressed in terms of the original variable to be a complete solution. Neglecting this step is a common mistake that can lead to an incorrect final answer.

Now that we have the final solution, it's always a good practice to check our work. We can do this by differentiating our answer and verifying that it matches the original integrand. If we differentiate (5/8)(x² + 2)⁴ + C with respect to x, we should get 5x(x²+2)³. Differentiation is the reverse process of integration, so it serves as a powerful tool for verifying the correctness of our integral.

In the next section, we'll discuss how to verify the solution and provide a summary of the steps involved in u-substitution.

Verification and Conclusion

To ensure the accuracy of our solution, we can verify it by differentiating the result we obtained, (5/8)(x² + 2)⁴ + C, and checking if it matches the original integrand, 5x(x²+2)³. Differentiation is the reverse operation of integration, so it provides a reliable way to confirm our work.

Let's differentiate (5/8)(x² + 2)⁴ + C with respect to x using the chain rule:

d/dx [(5/8)(x² + 2)⁴ + C] = (5/8) * 4(x² + 2)³ * (2x) + 0

Simplifying the expression, we get:

(5/2)(x² + 2)³ * (2x) = 5x(x² + 2)³

As we can see, the result of differentiation matches our original integrand, 5x(x²+2)³. This confirms that our integration was performed correctly, and our solution, (5/8)(x² + 2)⁴ + C, is indeed the correct antiderivative.

In conclusion, we have successfully computed the integral ∫5x(x²+2)³ dx using the u-substitution technique. The key steps involved were:

Identifying 'u': Choosing u = x² + 2 based on the composite function structure.
Computing 'du': Finding du = 2x dx and manipulating it to match the integrand.
Rewriting the integral: Expressing the integral in terms of 'u', resulting in (5/2) ∫u³ du.
Integrating with respect to 'u': Applying the power rule to get (5/8)u⁴ + C.
Substituting back to 'x': Replacing 'u' with x² + 2 to obtain the final answer, (5/8)(x² + 2)⁴ + C.
Verification: Differentiating the result to confirm it matches the original integrand.

The u-substitution technique is a powerful tool in integral calculus, allowing us to simplify complex integrals by transforming them into more manageable forms. By carefully choosing 'u' and manipulating 'du', we can often reduce an integral to a basic integration rule. This method is particularly effective when dealing with composite functions where the derivative of the inner function is present (or nearly present) in the integrand. Mastering u-substitution is essential for anyone studying calculus and its applications in various fields.

For further exploration and practice on integration techniques, consider visiting resources like Khan Academy's Calculus section.