Evaluating The Integral Of 4cos(3x) Dx A Step-by-Step Guide
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In the realm of calculus, evaluating integrals stands as a fundamental operation, underpinning numerous applications in physics, engineering, and economics. Among the diverse landscape of integrals, standard integrals involving trigonometric functions frequently emerge. This article delves into the evaluation of a specific standard integral: ∫4cos(3x) dx. We will embark on a step-by-step journey, unraveling the intricacies of this integral and equipping you with the knowledge to tackle similar challenges.
Understanding the Integral
Before we plunge into the evaluation process, let's first dissect the integral at hand. The integral ∫4cos(3x) dx represents the antiderivative of the function 4cos(3x). In essence, we seek a function whose derivative is 4cos(3x). The integrand, 4cos(3x), is a trigonometric function, specifically a cosine function with a scaled argument (3x) and a constant multiplier (4). The 'dx' signifies that we are integrating with respect to the variable x.
To effectively tackle this integral, we'll leverage our understanding of trigonometric integrals and employ a technique known as substitution. This method allows us to simplify the integral by introducing a new variable, making it more amenable to direct integration. With a solid grasp of the integral's components, we're primed to embark on the evaluation process, systematically unraveling its solution.
Trigonometric Integrals: A Foundation
Trigonometric integrals form a cornerstone of integral calculus, encompassing integrals involving trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant. These integrals frequently arise in various scientific and engineering contexts, underscoring the importance of mastering their evaluation. Among the fundamental trigonometric integrals are those involving sine and cosine, which serve as building blocks for more complex integrals.
The integrals of sine and cosine are elegantly straightforward: ∫sin(x) dx = -cos(x) + C and ∫cos(x) dx = sin(x) + C, where 'C' denotes the constant of integration. These foundational results stem directly from the derivatives of cosine and sine, respectively. However, when dealing with trigonometric integrals involving scaled arguments, such as cos(3x) in our case, a slight modification is required. This is where the technique of substitution comes into play, allowing us to transform the integral into a more manageable form.
Substitution, in essence, involves introducing a new variable to simplify the integrand. By carefully choosing the substitution, we can often reduce the integral to a standard form that can be readily evaluated. In the context of ∫4cos(3x) dx, the scaled argument '3x' hints at a suitable substitution. By letting u = 3x, we can express the integral in terms of 'u', making it easier to apply the fundamental integral of cosine.
The Power of Substitution: A Transformation Technique
Substitution, also known as u-substitution, stands as a cornerstone technique in integral calculus, empowering us to tackle integrals that initially appear daunting. This method hinges on the chain rule of differentiation in reverse, allowing us to transform integrals into simpler, more recognizable forms. The essence of substitution lies in identifying a suitable function within the integrand whose derivative also appears, or can be made to appear, in the integral.
The core idea behind substitution is to introduce a new variable, typically denoted as 'u', to represent a portion of the integrand. This substitution aims to simplify the integral by expressing it in terms of 'u' and 'du', where 'du' is the differential of 'u'. The success of substitution hinges on choosing the right substitution, one that effectively reduces the complexity of the integral. In many cases, the choice of 'u' is guided by the structure of the integrand, often involving a function and its derivative.
For instance, in our integral ∫4cos(3x) dx, the scaled argument '3x' within the cosine function suggests a potential substitution. By letting u = 3x, we establish a relationship between 'x' and 'u'. The next step involves finding the differential 'du' in terms of 'dx'. Differentiating both sides of u = 3x with respect to 'x', we obtain du/dx = 3, which implies du = 3dx. This relationship between 'du' and 'dx' is crucial for transforming the integral from 'x' to 'u'.
Step-by-Step Evaluation
Now, let's embark on the step-by-step evaluation of the integral ∫4cos(3x) dx, employing the substitution technique we've discussed.
Step 1: Identify the Substitution
As hinted earlier, the scaled argument '3x' within the cosine function suggests a suitable substitution. We let:
u = 3x
This substitution aims to simplify the integral by replacing '3x' with the single variable 'u'.
Step 2: Find the Differential
Next, we need to find the differential 'du' in terms of 'dx'. Differentiating both sides of u = 3x with respect to 'x', we get:
du/dx = 3
Multiplying both sides by 'dx', we obtain:
du = 3 dx
However, our original integral involves 'dx', not '3 dx'. To address this, we divide both sides of the equation by 3:
(1/3) du = dx
This equation expresses 'dx' in terms of 'du', which is essential for transforming the integral.
Step 3: Rewrite the Integral
Now, we rewrite the original integral in terms of 'u' and 'du'. We have:
∫4cos(3x) dx = ∫4cos(u) (1/3) du
We've replaced '3x' with 'u' and 'dx' with '(1/3) du'.
Step 4: Simplify the Integral
We can simplify the integral by factoring out the constant multipliers:
∫4cos(u) (1/3) du = (4/3) ∫cos(u) du
The constant (4/3) can be moved outside the integral sign, making the integral easier to manage.
Step 5: Evaluate the Integral
Now, we have a standard integral of cosine:
(4/3) ∫cos(u) du
The integral of cos(u) is sin(u) + C, where 'C' is the constant of integration. Therefore:
(4/3) ∫cos(u) du = (4/3) sin(u) + C
Step 6: Substitute Back
Our final step is to substitute back the original variable 'x'. Recall that we defined u = 3x. Substituting this back into the result, we get:
(4/3) sin(u) + C = (4/3) sin(3x) + C
Thus, the integral of 4cos(3x) dx is (4/3) sin(3x) + C.
The Final Result
After meticulously navigating the steps of substitution and integration, we arrive at the final result:
∫4cos(3x) dx = (4/3) sin(3x) + C
This equation represents the antiderivative of 4cos(3x). It signifies that if we differentiate (4/3) sin(3x) + C with respect to 'x', we will obtain 4cos(3x). The constant of integration, 'C', underscores the fact that there are infinitely many functions whose derivative is 4cos(3x), differing only by a constant value. This constant arises because the derivative of a constant is always zero.
In summary, we have successfully evaluated the standard integral ∫4cos(3x) dx by employing the technique of substitution. This method involved introducing a new variable 'u' to simplify the integral, evaluating the resulting integral in terms of 'u', and then substituting back to express the answer in terms of the original variable 'x'. This process highlights the power of substitution in tackling a wide range of integrals, particularly those involving composite functions. By mastering this technique, you'll be well-equipped to conquer a multitude of integration challenges.
Conclusion
Evaluating integrals is a cornerstone of calculus, and mastering techniques like substitution is crucial for tackling a wide array of problems. In this article, we have meticulously evaluated the standard integral ∫4cos(3x) dx, demonstrating the power and elegance of substitution. By breaking down the process into manageable steps, we've unraveled the intricacies of this integral and provided a clear roadmap for solving similar problems. Remember, practice is key to mastering integration techniques, so continue to explore and challenge yourself with various integrals. With dedication and the right tools, you'll become proficient in the art of integration.
