Solving ∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) Dx

This article delves into the step-by-step solution of the integral ∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) dx. This integral, at first glance, might seem daunting, but with the right techniques, it can be solved elegantly. We will break down the process into manageable steps, making it easy to understand even if you're not a seasoned mathematician. This is a fascinating example of how substitution and understanding the chain rule in reverse can simplify complex integrals. Let’s dive in and explore the solution together. In this comprehensive guide, we will not only solve the integral but also explain the underlying concepts and techniques used, ensuring that you grasp the methodology for similar problems. The integration process involves recognizing patterns and strategically applying methods to simplify the expression. Our focus will be on making the complex understandable, so you can tackle similar integration challenges with confidence. Prepare to enhance your mathematical toolkit with this detailed walkthrough.

Understanding the Integral

Before we jump into the solution, let's understand the integral we are dealing with. We have:

∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) dx

This integral involves trigonometric functions raised to various powers, and it includes a product of expressions. The presence of cos(x) alongside the sine functions hints at a possible substitution. When you first encounter an integral like this, it’s crucial to observe the components carefully. Notice how sin(x) and cos(x) are related through differentiation, which often suggests a u-substitution might be effective. Recognizing these patterns early on can save significant time and effort. The term (sin⁴(x) - 4sin³(x))⁶ is the most complex part, indicating it could be a suitable choice for substitution. The goal is to simplify the integral into a form that is easier to manage and solve using standard integration techniques. By breaking down the integral into its key components, we can develop a strategic approach to solving it. This initial analysis is a critical step in tackling any complex integral problem.

The U-Substitution Technique

The key to solving this integral lies in the u-substitution technique. This method simplifies integrals by replacing a complex expression with a single variable, making the integration process more straightforward. The success of u-substitution depends on identifying a suitable substitution that simplifies the integral while also accounting for its derivative. In this case, we notice that the derivative of the expression inside the parentheses in the power of 6 is closely related to the other terms in the integral. This is a telltale sign that u-substitution will be effective. Let's define our u as follows:

u = sin⁴(x) - 4sin³(x)

This substitution choice is strategic because the derivative of u will likely include terms involving sin(x) and cos(x), which are present in the integral. The goal here is to transform the integral into a simpler form in terms of u, allowing us to apply basic integration rules. The beauty of u-substitution is its ability to convert complex integrals into recognizable forms. By choosing the right expression for u, we can drastically reduce the complexity and solve the integral more efficiently. This technique is a cornerstone of integral calculus and is widely applicable across various types of integrals.

Finding du

Now that we have chosen our u, we need to find its derivative, du. This step is crucial because du will replace dx in the integral, allowing us to perform the integration with respect to u. Let's differentiate u with respect to x:

du/dx = d/dx (sin⁴(x) - 4sin³(x))

Using the chain rule, we get:

du/dx = 4sin³(x)cos(x) - 12sin²(x)cos(x)

We can factor out 4sin²(x)cos(x) from the expression:

du/dx = 4sin²(x)cos(x)(sin(x) - 3)

Now, we express du in terms of dx:

du = 4(sin³(x) - 3sin²(x))cos(x) dx

Notice that the expression (sin³(x) - 3sin²(x))cos(x) dx is present in our original integral, which is a great sign! This confirms that our choice of u was appropriate. The factor of 4 can be easily adjusted for. The key here is to accurately compute the derivative and manipulate it to match a part of the original integral. Finding du is a pivotal step in u-substitution, as it bridges the gap between the original integral and the simplified integral in terms of u. Careful differentiation and algebraic manipulation are essential for this step.

Adjusting the Integral

We have du = 4(sin³(x) - 3sin²(x))cos(x) dx, but in our integral, we have (sin³(x) - 3sin²(x))cos(x) dx. To make the substitution, we need to adjust our du to match this. We can do this by dividing both sides of the equation by 4:

(1/4)du = (sin³(x) - 3sin²(x))cos(x) dx

Now we can substitute u and (1/4)du into the original integral. This is where the magic of u-substitution happens. By correctly adjusting the differential, we can rewrite the integral in a much simpler form. The careful manipulation of du is crucial to ensure the substitution is accurate and the integral is properly transformed. This step highlights the importance of algebraic precision in calculus. We are essentially changing the variable of integration, and this adjustment ensures the integrity of the mathematical expression. Once the substitution is correctly set up, the integral becomes significantly easier to solve.

Performing the Substitution

Now, let's substitute u and (1/4)du into the original integral:

∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) dx = ∫(1/4)u⁶ du

This substitution has transformed the complex integral into a simple power rule integral. This is the goal of u-substitution: to simplify the integral into a form that can be easily integrated using standard rules. The expression ∫(1/4)u⁶ du is much more manageable than the original integral. By replacing the trigonometric functions with a single variable, we have made the problem significantly easier to solve. The substitution process is a powerful technique that showcases the elegance of calculus. It allows us to tackle complex problems by breaking them down into simpler components. With the integral now in a straightforward form, we can proceed to apply the power rule of integration.

Integrating with Respect to u

Now we integrate (1/4)u⁶ with respect to u. Using the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration, we get:

∫(1/4)u⁶ du = (1/4) ∫ u⁶ du

Applying the power rule:

(1/4) ∫ u⁶ du = (1/4) * (u⁷/7) + C

Simplifying, we have:

(1/4) * (u⁷/7) + C = u⁷/28 + C

So, the integral of (1/4)u⁶ with respect to u is u⁷/28 + C. This is a significant step, as we have successfully integrated the simplified expression. The power rule is a fundamental tool in integration, and its application here demonstrates the effectiveness of the u-substitution method. The constant of integration, C, is crucial as it represents the family of functions that have the same derivative. We are now one step closer to the final solution, but we must not forget to substitute back in terms of x.

Substituting Back

We have found the integral in terms of u, but we need to express the final answer in terms of x. To do this, we substitute back our original expression for u, which was:

u = sin⁴(x) - 4sin³(x)

Replacing u in our result u⁷/28 + C, we get:

(sin⁴(x) - 4sin³(x))⁷/28 + C

This is the final result of our integration. The process of substituting back is essential to ensure the solution is in the original variable. This step completes the integration process, giving us the antiderivative of the original function. The constant of integration, C, reminds us that there are infinitely many antiderivatives, differing only by a constant. The final expression represents the general solution to the integral. We have successfully navigated through the substitution, integration, and back-substitution steps to arrive at the answer. This comprehensive approach highlights the importance of each step in solving complex integrals.

The Final Solution

Therefore, the solution to the integral ∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) dx is:

(sin⁴(x) - 4sin³(x))⁷/28 + C

Where C is the constant of integration. This final answer encapsulates the entire process we've undertaken, from the initial substitution to the final back-substitution. The constant of integration is a critical part of the solution, as it acknowledges the family of functions that satisfy the integral. We have successfully solved a complex integral by strategically applying the u-substitution technique and the power rule of integration. This result not only provides the solution but also reinforces the understanding of integral calculus principles. The journey from the initial complex integral to this simplified solution showcases the power and elegance of mathematical techniques. By breaking down the problem into manageable steps, we were able to navigate through the intricacies and arrive at the correct answer.

Conclusion

In conclusion, we have successfully solved the integral ∫(sin³(x) - 3sin²(x))(sin⁴(x) - 4sin³(x))⁶cos(x) dx using the u-substitution technique. This method allowed us to transform a complex integral into a simpler form, making it easier to integrate. We identified a suitable substitution, computed its derivative, adjusted the integral, performed the integration, and then substituted back to obtain the final solution in terms of x. The final answer is (sin⁴(x) - 4sin³(x))⁷/28 + C, where C is the constant of integration. The power of u-substitution lies in its ability to simplify integrals by changing the variable of integration. This technique is widely applicable in calculus and is an essential tool for solving various types of integrals. The detailed walkthrough provided in this article demonstrates the step-by-step process, ensuring a clear understanding of the method. By mastering u-substitution, you can tackle a wide range of integration problems with confidence. Remember to always look for patterns and relationships within the integral to identify the most effective substitution. Practice is key to mastering this technique and building your calculus skills. We hope this explanation has been helpful and has enhanced your understanding of integral calculus.

For further learning on integration techniques, consider exploring resources like Khan Academy's Calculus section.