Evaluating The Integral Of E^(6θ)sin(7θ) A Step-by-Step Guide
This article delves into the intricate process of evaluating the integral ∫e^(6θ)sin(7θ) dθ. This type of integral, involving the product of an exponential function and a trigonometric function, often requires a technique called integration by parts. We will explore this technique in detail and apply it systematically to arrive at the solution, remembering to include the constant of integration.
Understanding Integration by Parts
Integration by parts is a powerful technique derived from the product rule of differentiation. It allows us to integrate products of functions by transforming the integral into a simpler form. The formula for integration by parts is:
∫u dv = uv - ∫v du
Where:
- u is a function of θ
- dv is the differential of another function of θ
- du is the differential of u
- v is the integral of dv
The key to successfully applying integration by parts lies in choosing appropriate functions for u and dv. The goal is to select u such that its derivative, du, simplifies the integral. Often, we use the acronym LIATE or ILATE as a guide for choosing u:
- L: Logarithmic functions
- I: Inverse trigonometric functions
- A: Algebraic functions
- T: Trigonometric functions
- E: Exponential functions
The function that appears earlier in the list is usually a good choice for u. This is a general guideline, and sometimes you may need to experiment with different choices to find the most effective approach. Understanding the underlying principles of integration by parts is crucial for effectively tackling complex integrals. We must carefully select the functions for u and dv, aiming to simplify the integral in subsequent steps. This strategic selection is paramount in navigating the complexities of integration and arriving at the correct solution. Mastering this technique expands our capacity to solve a broader range of integrals, particularly those involving products of different types of functions. The elegance of integration by parts lies in its ability to transform complex integrals into manageable forms, highlighting the interconnectedness of differentiation and integration in calculus. This method not only provides a solution but also deepens our appreciation for the mathematical tools available to us. Furthermore, integration by parts is not merely a formula to be memorized; it is a concept to be understood. By grasping the underlying principles, we can adapt the technique to various scenarios and confidently approach a wide array of integration problems. The strategic choice of u and dv is more than just a mechanical step; it is an exercise in mathematical intuition and problem-solving acumen.
Applying Integration by Parts to ∫e^(6θ)sin(7θ) dθ
In our integral, ∫e^(6θ)sin(7θ) dθ, we have an exponential function (e^(6θ)) and a trigonometric function (sin(7θ)). According to LIATE, we should choose sin(7θ) as u and e^(6θ)dθ as dv.
Let:
- u = sin(7θ)
- dv = e^(6θ) dθ
Then:
- du = 7cos(7θ) dθ
- v = ∫e^(6θ) dθ = (1/6)e^(6θ)
Now, applying the integration by parts formula:
∫e^(6θ)sin(7θ) dθ = sin(7θ) * (1/6)e^(6θ) - ∫(1/6)e^(6θ) * 7cos(7θ) dθ
Simplifying, we get:
∫e^(6θ)sin(7θ) dθ = (1/6)e^(6θ)sin(7θ) - (7/6)∫e^(6θ)cos(7θ) dθ
Notice that we now have another integral, ∫e^(6θ)cos(7θ) dθ, which also requires integration by parts. We will apply the technique again, choosing u and dv similarly.
Let:
- u = cos(7θ)
- dv = e^(6θ) dθ
Then:
- du = -7sin(7θ) dθ
- v = (1/6)e^(6θ)
Applying integration by parts again:
∫e^(6θ)cos(7θ) dθ = cos(7θ) * (1/6)e^(6θ) - ∫(1/6)e^(6θ) * (-7sin(7θ)) dθ
Simplifying:
∫e^(6θ)cos(7θ) dθ = (1/6)e^(6θ)cos(7θ) + (7/6)∫e^(6θ)sin(7θ) dθ
Now, we substitute this result back into our original equation:
∫e^(6θ)sin(7θ) dθ = (1/6)e^(6θ)sin(7θ) - (7/6)[(1/6)e^(6θ)cos(7θ) + (7/6)∫e^(6θ)sin(7θ) dθ]
This step highlights the iterative nature of integration by parts when dealing with products of exponential and trigonometric functions. The repeated application of the technique, although initially seeming to complicate the expression, ultimately leads to a solvable equation. Each iteration carefully peels away a layer of complexity, revealing the underlying structure of the integral. The strategic choice of u and dv in each step is crucial, ensuring that the integral gradually transforms into a manageable form. This process underscores the importance of meticulousness and precision in mathematical manipulations. By systematically applying integration by parts, we navigate the intricacies of the integral and pave the way for a final solution. The re-emergence of the original integral in the equation is a key observation, allowing us to treat the entire expression algebraically and solve for the unknown integral. This clever maneuver demonstrates the power of mathematical reasoning and the ability to transform a seemingly intractable problem into a solvable one. The journey through repeated integration by parts is a testament to the elegance and effectiveness of this fundamental calculus technique. It showcases the interconnectedness of different mathematical concepts and the importance of a systematic approach to problem-solving.
Solving for the Integral
Expanding and rearranging the equation:
∫e^(6θ)sin(7θ) dθ = (1/6)e^(6θ)sin(7θ) - (7/36)e^(6θ)cos(7θ) - (49/36)∫e^(6θ)sin(7θ) dθ
Now, we can add (49/36)∫e^(6θ)sin(7θ) dθ to both sides:
(1 + 49/36)∫e^(6θ)sin(7θ) dθ = (1/6)e^(6θ)sin(7θ) - (7/36)e^(6θ)cos(7θ)
(85/36)∫e^(6θ)sin(7θ) dθ = (1/6)e^(6θ)sin(7θ) - (7/36)e^(6θ)cos(7θ)
Multiply both sides by (36/85):
∫e^(6θ)sin(7θ) dθ = (36/85)[(1/6)e^(6θ)sin(7θ) - (7/36)e^(6θ)cos(7θ)]
Simplifying:
∫e^(6θ)sin(7θ) dθ = (6/85)e^(6θ)sin(7θ) - (7/85)e^(6θ)cos(7θ)
Finally, we add the constant of integration, C:
∫e^(6θ)sin(7θ) dθ = (6/85)e^(6θ)sin(7θ) - (7/85)e^(6θ)cos(7θ) + C
This algebraic manipulation is a critical step in solving for the integral. By carefully rearranging the equation, we isolate the desired integral on one side, paving the way for a final solution. The addition and multiplication operations are performed with meticulous attention to detail, ensuring accuracy in the result. This process highlights the power of algebraic techniques in unraveling complex mathematical expressions. Each step is a deliberate maneuver, transforming the equation into a more manageable form. The constant of integration, C, is a crucial addition, acknowledging the family of functions that have the same derivative. This underscores the fundamental principles of integral calculus and the importance of complete and accurate solutions. The final expression is a testament to the effectiveness of integration by parts and the power of algebraic manipulation. It represents a complete and accurate solution to the original integral, demonstrating our understanding of calculus principles. Furthermore, this process exemplifies the iterative nature of problem-solving in mathematics. By systematically applying techniques and carefully manipulating expressions, we can navigate complex challenges and arrive at elegant solutions. The satisfaction of obtaining a final result after a series of steps is a rewarding experience, reinforcing our appreciation for the beauty and power of mathematics.
The Final Solution
Therefore, the integral of e^(6θ)sin(7θ) dθ is:
∫e^(6θ)sin(7θ) dθ = (6/85)e^(6θ)sin(7θ) - (7/85)e^(6θ)cos(7θ) + C
Where C is the constant of integration.
This final solution encapsulates the entire process of evaluating the integral, from the initial application of integration by parts to the algebraic manipulation and the inclusion of the constant of integration. It represents a complete and accurate answer, demonstrating our mastery of calculus techniques. The expression itself is a testament to the elegance of mathematics, a concise representation of a complex process. Each term in the solution plays a crucial role, reflecting the interplay between the exponential and trigonometric functions in the original integral. The constant of integration, C, reminds us of the broader context of integration, the family of functions that share the same derivative. This comprehensive solution not only provides an answer but also reinforces our understanding of the underlying principles of calculus. It is a culmination of careful reasoning, meticulous calculations, and a systematic approach to problem-solving. The satisfaction of arriving at this final result is a testament to the power of mathematical tools and our ability to apply them effectively. Furthermore, this solution serves as a foundation for further exploration of calculus concepts and their applications in various fields. The journey from the initial integral to the final solution is a testament to the beauty and power of mathematics, a process that combines creativity, logic, and precision.
Conclusion
Evaluating integrals like ∫e^(6θ)sin(7θ) dθ requires a solid understanding of integration techniques, particularly integration by parts. By carefully choosing u and dv and applying the formula iteratively, we can successfully solve these integrals. Remember to always include the constant of integration in the final answer. This process showcases the power and elegance of calculus in solving complex mathematical problems. The key takeaway is the importance of a systematic approach and meticulous attention to detail in navigating the intricacies of integration. Mastering integration by parts opens doors to solving a broader range of integrals, empowering us to tackle more complex mathematical challenges. This technique is not merely a formula to be memorized; it is a tool to be understood and applied with precision and strategic thinking. The beauty of mathematics lies in its ability to transform complex problems into manageable steps, ultimately leading to elegant and satisfying solutions. Furthermore, the constant of integration serves as a reminder of the richness and completeness of the solution, acknowledging the family of functions that satisfy the integral. The journey through this process enhances our mathematical intuition and strengthens our problem-solving skills, preparing us for future challenges in calculus and beyond. The ability to confidently apply integration by parts is a valuable asset in any mathematical endeavor, demonstrating a deep understanding of calculus principles and a commitment to accuracy and precision.
