Solving The Integral Of X³(x⁴-2)³⁸ Dx Using Substitution

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In this article, we will delve into the process of evaluating the integral ∫x³(x⁴-2)³⁸ dx. This type of integral is best solved using the substitution method, a powerful technique in calculus that simplifies integration by introducing a new variable. We will walk through each step, making the substitution u = x⁴ - 2, and ultimately express the solution in terms of x. Understanding and mastering the substitution method is crucial for tackling a wide range of integration problems, especially those involving composite functions. So, let's embark on this mathematical journey and unravel the solution together.

Understanding the Substitution Method

The substitution method, also known as u-substitution, is a technique used to simplify integrals by replacing a complex expression within the integrand with a single variable, typically u. The core idea behind this method is to reverse the chain rule of differentiation. When we differentiate a composite function, we apply the chain rule, which involves differentiating the outer function and then multiplying by the derivative of the inner function. In integration, we aim to undo this process. By carefully choosing a substitution, we can often transform a complicated integral into a simpler one that we can readily evaluate. The key is to identify a suitable substitution that simplifies the integrand and its differential. This often involves recognizing a function and its derivative (or a multiple thereof) within the integral. For instance, if we have an integral of the form ∫f(g(x))g'(x) dx, we can substitute u = g(x), which gives us du = g'(x) dx. The integral then becomes ∫f(u) du, which might be easier to evaluate. After finding the integral in terms of u, we need to substitute back g(x) for u to express the final answer in terms of the original variable x. The substitution method is a versatile tool that can be applied to various types of integrals, including those involving trigonometric functions, exponential functions, and algebraic functions. Its effectiveness lies in its ability to transform complex integrals into simpler forms, making the integration process more manageable and efficient.

Step-by-Step Solution

1. Identify the Substitution

The first crucial step in solving this integral using substitution is to identify a suitable expression to replace with a new variable, u. Looking at the integral ∫x³(x⁴-2)³⁸ dx, we observe that the expression (x⁴ - 2) appears within the parentheses, raised to the power of 38. This suggests that making the substitution u = x⁴ - 2 might simplify the integral. The reason for this choice becomes clear when we consider the derivative of u with respect to x. Differentiating u = x⁴ - 2 with respect to x, we get du/dx = 4x³. Notice that x³ is also present in the original integral, which is a strong indication that this substitution is a good choice. By substituting u = x⁴ - 2, we are essentially simplifying the complex part of the integrand, the term raised to the power of 38. This substitution will allow us to transform the integral into a more manageable form. The ability to recognize such patterns and choose appropriate substitutions is a key skill in integral calculus. A well-chosen substitution can significantly simplify the integration process, making it easier to find the antiderivative.

2. Find du

Once we've identified the substitution u = x⁴ - 2, the next step is to find du, which represents the differential of u. This involves differentiating u with respect to x and then expressing the result in terms of du and dx. As we found in the previous step, differentiating u = x⁴ - 2 with respect to x gives us du/dx = 4x³. Now, we need to isolate du to express it in terms of dx. Multiplying both sides of the equation by dx, we get du = 4x³ dx. This equation relates the differential of u to the differential of x. However, if we observe the original integral ∫x³(x⁴-2)³⁸ dx, we only have x³ dx, not 4x³ dx. Therefore, we need to manipulate the equation du = 4x³ dx to match the expression in the integral. We can do this by dividing both sides of the equation by 4, which gives us (1/4) du = x³ dx. This equation is crucial because it allows us to replace x³ dx in the original integral with (1/4) du. Finding du correctly is essential for the substitution method to work effectively. It ensures that we can transform the integral entirely in terms of the new variable u, without any lingering x terms. This step bridges the gap between the original variable x and the new variable u, enabling us to simplify the integral and proceed with the integration process.

3. Substitute into the Integral

Now that we have both u and du in terms of x, we can substitute these expressions into the original integral. Recall that we have u = x⁴ - 2 and (1/4) du = x³ dx. Substituting these into the integral ∫x³(x⁴-2)³⁸ dx, we get ∫(x⁴-2)³⁸ * x³ dx = ∫u³⁸ * (1/4) du. This substitution has transformed the integral from an expression involving powers of x to a much simpler expression involving only u. The factor of (1/4) is a constant and can be moved outside the integral, further simplifying the expression. This gives us (1/4)∫u³⁸ du. This new integral is significantly easier to evaluate than the original one. It involves a simple power function, which we can integrate using the power rule for integration. The key to the substitution method is this ability to transform a complex integral into a simpler form through a clever change of variables. By substituting u and du, we have effectively rewritten the integral in terms of a new variable that makes the integration process more straightforward. This step highlights the power of the substitution method in simplifying integrals and making them more accessible to standard integration techniques.

4. Integrate with Respect to u

After the substitution, we have the simplified integral (1/4)∫u³⁸ du. To evaluate this integral, we can use the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n is any real number except -1, and C is the constant of integration. Applying this rule to our integral, we have (1/4)∫u³⁸ du = (1/4) * (u³⁹/39) + C. This step involves finding the antiderivative of u³⁸, which is u³⁹/39. We also multiply by the constant factor (1/4) that was outside the integral. The constant of integration, C, is added because the derivative of a constant is zero, so any constant could be added to the antiderivative and still satisfy the original integral. Simplifying the expression, we get (1/4) * (u³⁹/39) + C = u³⁹/156 + C. This is the integral in terms of u. However, the original problem was given in terms of x, so we need to substitute back to express our final answer in terms of x. Integrating with respect to u is a crucial step in the substitution method. It allows us to find the antiderivative of the simplified expression, which is often much easier to compute than the antiderivative of the original integrand. The power rule is a fundamental tool in integration, and its application here demonstrates its versatility in handling power functions. Remember to always include the constant of integration, C, when finding indefinite integrals, as it represents the family of functions that have the same derivative.

5. Substitute Back for x

Having found the integral in terms of u, the final step is to substitute back the original expression for u in terms of x. Recall that we made the substitution u = x⁴ - 2. Now, we replace u with (x⁴ - 2) in our result, which was u³⁹/156 + C. This gives us (x⁴ - 2)³⁹/156 + C. This is the final answer, expressed in terms of the original variable x. We have successfully evaluated the integral ∫x³(x⁴-2)³⁸ dx using the substitution method. The key to this step is to remember the original substitution and reverse it to express the antiderivative in terms of the original variable. This ensures that our solution is in the same context as the original problem. Substituting back for x completes the integration process and provides the final answer to the problem. The constant of integration, C, remains in the final answer, indicating that there are infinitely many functions that have the same derivative as the original integrand. This step emphasizes the importance of understanding the relationship between the original and substituted variables and the need to express the final answer in the appropriate context.

Final Answer

Therefore, the integral ∫x³(x⁴-2)³⁸ dx evaluates to (x⁴ - 2)³⁹/156 + C. This result is obtained by carefully applying the substitution method, which involves identifying a suitable substitution, finding the differential, substituting into the integral, integrating with respect to the new variable, and finally, substituting back to express the answer in terms of the original variable. The substitution method is a powerful technique for simplifying integrals, and this example demonstrates its effectiveness in handling integrals involving composite functions. Mastering this method is essential for anyone studying calculus and its applications. The final answer represents the antiderivative of the original integrand, and the constant of integration, C, indicates the family of functions that have the same derivative. This result provides a complete solution to the problem and showcases the elegance and power of the substitution method in integral calculus.

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Evaluate the integral ∫x³(x⁴-2)³⁸ dx using the substitution u = x⁴ - 2. Express your answer in terms of x and include the constant of integration, C.

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Solving ∫x³(x⁴-2)³⁸ dx A Step-by-Step Guide with Substitution