Evaluate The Definite Integral Of Cos^6(3x) From 0 To Π/6. How To Solve The Integral Of Cos^6(3x)?
In this article, we will delve into the process of evaluating the definite integral of cos^6(3x) with respect to x, over the interval from 0 to π/6. This integral falls under the category of trigonometric integrals, which often require the use of trigonometric identities and reduction formulas to simplify the integrand. Our main keyword is integral of cos^6(3x), and we'll explore various techniques to effectively solve this problem. The integration of trigonometric functions is a fundamental concept in calculus, with applications spanning across physics, engineering, and other scientific disciplines. Mastering these techniques is essential for anyone looking to deepen their understanding of calculus and its applications. This particular integral presents a good challenge due to the power of the cosine function, necessitating the use of power-reduction formulas to simplify the expression before integration. Let's embark on this mathematical journey, breaking down each step to arrive at the solution. The journey of solving this integral involves a meticulous application of trigonometric identities and integral calculus techniques. We aim to provide a comprehensive explanation that not only solves the problem but also enhances the reader's understanding of the underlying mathematical principles. So, without further ado, let's begin our exploration of this fascinating integral and uncover the beauty of mathematical problem-solving.
Understanding the Problem
Before diving into the solution, it's crucial to understand the problem statement clearly. We are tasked with evaluating the definite integral:
∫[0 to π/6] cos^6(3x) dx
This integral represents the area under the curve of the function y = cos^6(3x) between the limits x = 0 and x = π/6. The integrand, cos^6(3x), is a trigonometric function raised to the sixth power, which can be challenging to integrate directly. Our primary goal is to simplify this expression using trigonometric identities and then apply the fundamental theorem of calculus to find the definite integral. The key to solving this integral lies in our ability to transform the integrand into a form that is easier to integrate. This involves using power-reduction formulas to express cos^6(3x) in terms of lower powers of cosine and sine. By breaking down the problem into smaller, manageable steps, we can systematically approach the solution. Understanding the behavior of the cosine function and its transformations is also crucial in this context. The function cos(3x) oscillates three times faster than the standard cos(x) function, which will affect the integration process. Furthermore, raising the cosine function to the sixth power introduces additional complexity, necessitating the use of trigonometric identities to simplify the expression. Our initial focus will be on applying the appropriate power-reduction formulas to reduce the power of the cosine function and make the integral more tractable.
Trigonometric Identities and Reduction Formulas
The core of solving this integral lies in the strategic application of trigonometric identities, specifically the power-reduction formulas. These formulas allow us to express higher powers of trigonometric functions in terms of lower powers, making integration significantly easier. The power-reduction formula for cosine is given by:
cos^2(θ) = (1 + cos(2θ))/2
However, we have cos^6(3x), which can be written as (cos2(3x))3. We will apply the power-reduction formula multiple times to reduce the power. First, we apply the formula to cos^2(3x):
cos^2(3x) = (1 + cos(6x))/2
Now, we need to cube this expression:
(cos2(3x))3 = [(1 + cos(6x))/2]^3 = (1 + cos(6x))^3 / 8
Expanding the cube, we get:
(1 + cos(6x))^3 = 1 + 3cos(6x) + 3cos^2(6x) + cos^3(6x)
Now, we need to deal with the cos^2(6x) and cos^3(6x) terms. We apply the power-reduction formula again to cos^2(6x):
cos^2(6x) = (1 + cos(12x))/2
For cos^3(6x), we can write it as cos(6x) * cos^2(6x) and then apply the power-reduction formula to cos^2(6x):
cos^3(6x) = cos(6x) * (1 + cos(12x))/2 = (cos(6x) + cos(6x)cos(12x))/2
Now, we use the product-to-sum formula for cos(6x)cos(12x):
cos(A)cos(B) = (1/2)[cos(A - B) + cos(A + B)]
cos(6x)cos(12x) = (1/2)[cos(-6x) + cos(18x)] = (1/2)[cos(6x) + cos(18x)]
So,
cos^3(6x) = (cos(6x) + (1/2)[cos(6x) + cos(18x)])/2 = (3cos(6x) + cos(18x))/4
By applying these trigonometric identities and reduction formulas, we've successfully transformed the original integrand into a more manageable form. The next step involves substituting these expressions back into the integral and performing the integration.
Simplifying the Integrand
Now that we have applied the trigonometric identities and reduction formulas, let's simplify the integrand. We have:
cos^6(3x) = (1 + 3cos(6x) + 3cos^2(6x) + cos^3(6x))/8
We found that:
cos^2(6x) = (1 + cos(12x))/2
cos^3(6x) = (3cos(6x) + cos(18x))/4
Substituting these back into the expression for cos^6(3x), we get:
cos^6(3x) = (1 + 3cos(6x) + 3(1 + cos(12x))/2 + (3cos(6x) + cos(18x))/4)/8
Multiplying through by the denominators to clear fractions, we have:
cos^6(3x) = (4 + 12cos(6x) + 6(1 + cos(12x)) + 3cos(6x) + cos(18x))/32
Simplifying further:
cos^6(3x) = (4 + 12cos(6x) + 6 + 6cos(12x) + 3cos(6x) + cos(18x))/32
Combining like terms:
cos^6(3x) = (10 + 15cos(6x) + 6cos(12x) + cos(18x))/32
Now, we have successfully simplified the integrand into a sum of cosine functions with different frequencies. This form is much easier to integrate term by term. The next step is to integrate this simplified expression over the given limits of integration.
Performing the Integration
With the integrand simplified, we can now perform the integration. We have:
∫[0 to π/6] cos^6(3x) dx = ∫[0 to π/6] (10 + 15cos(6x) + 6cos(12x) + cos(18x))/32 dx
We can integrate term by term:
∫[0 to π/6] cos^6(3x) dx = (1/32) ∫[0 to π/6] (10 + 15cos(6x) + 6cos(12x) + cos(18x)) dx
Now, we integrate each term:
∫ 10 dx = 10x ∫ 15cos(6x) dx = (15/6)sin(6x) = (5/2)sin(6x) ∫ 6cos(12x) dx = (6/12)sin(12x) = (1/2)sin(12x) ∫ cos(18x) dx = (1/18)sin(18x)
So, the integral becomes:
(1/32) [10x + (5/2)sin(6x) + (1/2)sin(12x) + (1/18)sin(18x)] evaluated from 0 to π/6
Now, we evaluate the expression at the limits of integration:
At x = π/6:
10(π/6) + (5/2)sin(6 * π/6) + (1/2)sin(12 * π/6) + (1/18)sin(18 * π/6) = (5π/3) + (5/2)sin(π) + (1/2)sin(2π) + (1/18)sin(3π) = 5π/3
At x = 0:
10(0) + (5/2)sin(0) + (1/2)sin(0) + (1/18)sin(0) = 0
So, the definite integral is:
(1/32) * (5π/3 - 0) = 5π/96
Thus, the value of the integral is 5π/96.
Final Answer
After meticulously applying trigonometric identities, reduction formulas, and integration techniques, we have successfully evaluated the definite integral:
∫[0 to π/6] cos^6(3x) dx = 5π/96
This result represents the exact area under the curve of the function y = cos^6(3x) between the limits x = 0 and x = π/6. The process involved transforming the integrand using power-reduction formulas, simplifying the expression, integrating term by term, and finally evaluating the definite integral. This exercise demonstrates the power of trigonometric identities and integral calculus in solving complex problems. The journey from the initial integral to the final answer showcases the beauty and elegance of mathematical problem-solving. By breaking down the problem into smaller, manageable steps, we were able to systematically approach the solution and arrive at the correct result. This integral serves as a valuable example for students and enthusiasts alike, highlighting the importance of mastering trigonometric identities and integration techniques. The final answer, 5π/96, is a testament to the power of mathematical tools in uncovering the hidden relationships and values within mathematical expressions. Through this exploration, we have not only solved the integral but also deepened our understanding of the underlying mathematical principles.