Evaluating ∫x³(x⁴-6)¹² Dx With Substitution A Step-by-Step Guide

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Evaluating integrals is a fundamental skill in calculus, and the technique of substitution is a powerful tool for simplifying complex integrals. In this article, we will delve into the process of evaluating the integral ∫x³(x⁴-6)¹² dx using the substitution method. This method allows us to transform a complicated integral into a simpler one by introducing a new variable, making the integration process more manageable. We will walk through each step of the process, ensuring a clear understanding of how to apply substitution effectively.

Understanding the Substitution Method

The substitution method, often referred to as u-substitution, is a technique used to simplify integrals by replacing a complex expression within the integral with a single variable, u. The key idea behind this method is to identify a function and its derivative within the integral. By making an appropriate substitution, we can often transform the integral into a more recognizable form that we can easily integrate. The general process involves the following steps:

  1. Identify a suitable substitution: Look for a function and its derivative (or a constant multiple of its derivative) within the integral.
  2. Let u equal the function: Define a new variable u as the function you've identified.
  3. Find du/dx: Calculate the derivative of u with respect to x.
  4. Solve for dx: Express dx in terms of du.
  5. Substitute: Replace the original function and dx in the integral with u and du, respectively.
  6. Evaluate the integral: Integrate the new integral with respect to u.
  7. Substitute back: Replace u with its original expression in terms of x.
  8. Add the constant of integration: Remember to add the constant of integration, C, since the antiderivative is not unique.

In our case, the integral we want to evaluate is ∫x³(x⁴-6)¹² dx. By carefully observing the integrand, we can see that the derivative of (x⁴-6) is related to x³, which suggests that substitution might be a suitable approach. Let's proceed with the steps of the substitution method to solve this integral.

Applying Substitution to ∫x³(x⁴-6)¹² dx

Step 1: Identify a Suitable Substitution

In the integral ∫x³(x⁴-6)¹² dx, we can observe that the expression (x⁴-6) is raised to the power of 12, making it a complex part of the integrand. The derivative of (x⁴-6) is 4x³, which is a multiple of x³ present in the integral. This observation suggests that letting u = x⁴-6 might simplify the integral. The presence of x³ as a factor makes this substitution particularly promising, as it can be readily incorporated into the du term.

Step 2: Let u = x⁴ - 6

We define our substitution by setting u equal to the expression inside the parentheses:

u = x⁴ - 6

This substitution aims to simplify the integrand by replacing the complex expression (x⁴-6) with a single variable, u. This is a crucial step in transforming the integral into a more manageable form. By choosing this substitution, we are essentially encapsulating the more complicated part of the integrand into a single variable, paving the way for easier integration.

Step 3: Find du/dx

Next, we need to find the derivative of u with respect to x, denoted as du/dx. This derivative will help us relate du and dx, which is essential for the substitution process. Differentiating both sides of the equation u = x⁴ - 6 with respect to x, we get:

du/dx = 4x³

The derivative du/dx provides the crucial link between u and x, allowing us to express dx in terms of du. This step is fundamental to changing the variable of integration from x to u. The derivative we obtained, 4x³, confirms our initial observation that the substitution is appropriate, as it includes the x³ term present in the original integral.

Step 4: Solve for dx

Now we need to isolate dx in terms of du. From the previous step, we have du/dx = 4x³. To solve for dx, we can rearrange the equation as follows:

du = 4x³ dx

Divide both sides by 4x³ to get:

dx = du / (4x³)

Expressing dx in terms of du is a key step in the substitution method. It allows us to replace dx in the original integral with an expression involving du, effectively changing the variable of integration. This manipulation is essential for transforming the integral into a form that is easier to integrate.

Step 5: Substitute

Now we substitute u and dx into the original integral. Recall that our original integral is ∫x³(x⁴-6)¹² dx. We replace (x⁴-6) with u and dx with du / (4x³). The integral becomes:

∫x³(u)¹² (du / (4x³))

Notice that the x³ terms cancel out, simplifying the integral significantly. This cancellation is a hallmark of a successful substitution, as it eliminates the original variable x from the integrand, leaving us with an integral solely in terms of u. After canceling x³, we have:

∫(u)¹² (du / 4)

We can rewrite this as:

(1/4) ∫u¹² du

The substitution has transformed the original complex integral into a much simpler one that is readily integrable.

Step 6: Evaluate the Integral

We can now evaluate the integral with respect to u. The power rule for integration states that ∫uⁿ du = (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Applying this rule to our integral, we get:

(1/4) ∫u¹² du = (1/4) * (u¹³ / 13) + C

Simplifying, we have:

(1/52)u¹³ + C

This is the antiderivative in terms of u. We have successfully integrated the transformed integral, but we are not yet done. The final step is to return to the original variable, x.

Step 7: Substitute Back

To express the result in terms of x, we need to substitute u back with its original expression, which is u = x⁴ - 6. Replacing u in our antiderivative, we get:

(1/52)(x⁴ - 6)¹³ + C

This is the antiderivative of the original integral, expressed in terms of x. We have successfully reversed the substitution, returning the integral to its original variable.

Step 8: Add the Constant of Integration

Finally, we include the constant of integration, C, since the antiderivative is a family of functions that differ by a constant. Our final result is:

(1/52)(x⁴ - 6)¹³ + C

This constant represents the ambiguity inherent in the antiderivative, as the derivative of a constant is always zero. Thus, we add C to acknowledge the infinite possible vertical shifts of the antiderivative function.

Final Answer

Therefore, the integral ∫x³(x⁴-6)¹² dx, evaluated using the substitution u = x⁴ - 6, is:

(1/52)(x⁴ - 6)¹³ + C

This comprehensive step-by-step solution demonstrates how the substitution method simplifies the integration process. By carefully choosing an appropriate substitution, we transformed a complex integral into a straightforward one, highlighting the power and utility of this technique in calculus. The ability to skillfully apply substitution is a valuable asset in solving a wide range of integration problems.

Conclusion

In conclusion, evaluating the integral ∫x³(x⁴-6)¹² dx using the substitution method demonstrates a powerful technique in calculus. By letting u = x⁴ - 6, we transformed the integral into a simpler form that was easily integrated. This process underscores the importance of recognizing patterns within integrals and strategically applying substitution to simplify the problem. The final result, (1/52)(x⁴ - 6)¹³ + C, showcases the effectiveness of the substitution method in solving complex integrals. Mastering this technique is crucial for anyone studying calculus and related fields, as it provides a versatile approach to handling a wide variety of integration challenges. The ability to identify suitable substitutions and execute the steps accurately is a testament to a strong understanding of calculus principles.