Finding The Antiderivative Of 8cos(x) - 5sin(x) A Step-by-Step Solution

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As we delve into the fascinating world of calculus, understanding antiderivatives is crucial. Antiderivatives, in essence, are the reverse operation of differentiation. This article will provide a step-by-step guide on how to find the general antiderivative of the function f(x) = 8cos(x) - 5sin(x), and more importantly, how to verify our answer through differentiation. This process highlights the fundamental relationship between differentiation and integration, two pillars of calculus.

Understanding Antiderivatives: The Reverse of Differentiation

Before we jump into the specific problem, let's solidify our understanding of antiderivatives. Think of differentiation as a process that breaks down a function into its rate of change. Antiderivatives, conversely, rebuild the original function from its rate of change. This 'rebuilding' isn't always perfect; there's often a constant term that disappears during differentiation, and we need to account for it when finding antiderivatives. This constant is denoted by "C" and is called the constant of integration.

To truly grasp the concept, let’s consider a simple example. Imagine we differentiate x². The result is 2x. Now, if we're given 2x and asked to find its antiderivative, we know x² is a possibility. But so is x² + 1, or x² - 5, or even x² + π. All these functions have the same derivative, 2x, because the derivative of any constant is zero. This is why we always include "C" when finding the general antiderivative, acknowledging the infinite possibilities for the constant term.

The power rule in differentiation states that the derivative of xⁿ is nxⁿ⁻¹. The antiderivative, in essence, reverses this. To find the antiderivative of xⁿ, we increase the power by 1 (to n+1) and divide by the new power (n+1). Of course, this rule has an exception: when n = -1. The antiderivative of x⁻¹ (or 1/x) is the natural logarithm, ln|x|. The absolute value is crucial here because the natural logarithm is only defined for positive values, while 1/x is defined for all non-zero values.

Trigonometric functions also have well-defined antiderivatives. The derivative of sin(x) is cos(x), so the antiderivative of cos(x) is sin(x) + C. Conversely, the derivative of cos(x) is -sin(x), which means the antiderivative of sin(x) is -cos(x) + C. These relationships are fundamental and should be memorized for efficient problem-solving. The key is to think about what function, when differentiated, will give you the function you're trying to find the antiderivative of.

Understanding the linearity of antiderivatives is another crucial aspect. This property states that the antiderivative of a sum (or difference) of functions is the sum (or difference) of their individual antiderivatives. Mathematically, this can be expressed as ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx. Furthermore, the antiderivative of a constant multiplied by a function is the constant multiplied by the antiderivative of the function: ∫[k * f(x)] dx = k * ∫f(x) dx, where k is a constant. These properties greatly simplify the process of finding antiderivatives of complex functions.

Step-by-Step Solution: Finding the Antiderivative

Now, let's tackle the problem at hand: finding the general antiderivative of f(x) = 8cos(x) - 5sin(x). We'll leverage the properties and rules we just discussed to systematically solve this problem.

1. Apply the Linearity Property:

The first step is to use the linearity property of antiderivatives. This allows us to break down the problem into smaller, more manageable parts. We can rewrite the integral of f(x) as the difference of two separate integrals:

∫[8cos(x) - 5sin(x)] dx = ∫8cos(x) dx - ∫5sin(x) dx

This step simplifies the problem by allowing us to focus on each term individually.

2. Factor out Constants:

Next, we can factor out the constant multiples from each integral. This is another application of the linearity property:

∫8cos(x) dx - ∫5sin(x) dx = 8∫cos(x) dx - 5∫sin(x) dx

By factoring out the constants, we further simplify the integration process. We now only need to find the antiderivatives of cos(x) and sin(x), which are standard results.

3. Find the Antiderivatives of cos(x) and sin(x):

Recall that the antiderivative of cos(x) is sin(x), and the antiderivative of sin(x) is -cos(x). These are fundamental trigonometric antiderivatives that are essential to remember.

So, we have:

8∫cos(x) dx = 8sin(x)

And:

5∫sin(x) dx = 5(-cos(x)) = -5cos(x)

4. Combine the Results:

Now, we substitute these antiderivatives back into our expression:

8sin(x) - (-5cos(x)) = 8sin(x) + 5cos(x)

5. Add the Constant of Integration:

The final and crucial step is to add the constant of integration, C. This accounts for the infinite possibilities of constant terms that could have disappeared during differentiation:

8sin(x) + 5cos(x) + C

Therefore, the general antiderivative of f(x) = 8cos(x) - 5sin(x) is F(x) = 8sin(x) + 5cos(x) + C.

Checking the Answer by Differentiating

The best way to ensure we've found the correct antiderivative is to differentiate our result and see if we get back our original function. This process highlights the inverse relationship between differentiation and integration.

1. Differentiate the Antiderivative:

We need to find the derivative of F(x) = 8sin(x) + 5cos(x) + C. We'll differentiate each term separately.

  • The derivative of 8sin(x) is 8cos(x) (since the derivative of sin(x) is cos(x)).
  • The derivative of 5cos(x) is -5sin(x) (since the derivative of cos(x) is -sin(x)).
  • The derivative of the constant C is 0.

2. Combine the Derivatives:

Adding these results together, we get:

8cos(x) - 5sin(x) + 0 = 8cos(x) - 5sin(x)

3. Compare with the Original Function:

This result, 8cos(x) - 5sin(x), is exactly our original function, f(x). This confirms that our antiderivative, F(x) = 8sin(x) + 5cos(x) + C, is correct.

Conclusion: Mastering Antiderivatives

Finding antiderivatives is a fundamental skill in calculus. By understanding the relationship between differentiation and integration, applying the linearity property, and remembering basic antiderivatives, we can effectively solve these problems. The key takeaway is to always check your answer by differentiating – this ensures accuracy and reinforces the core concepts. Practice is essential to mastering this skill. Work through various examples, gradually increasing the complexity, and you'll become proficient in finding antiderivatives. This will not only help you in your calculus studies but also provide a solid foundation for more advanced mathematical concepts.

In summary, we successfully found the general antiderivative of f(x) = 8cos(x) - 5sin(x), which is F(x) = 8sin(x) + 5cos(x) + C. We then rigorously verified our solution by differentiating F(x) and confirming that it returned our original function, f(x). This process exemplifies the power and beauty of calculus, where differentiation and integration work in harmony to solve complex problems. This skill is incredibly important and a great tool for any future mathematician. Always practice and always check your work to ensure that you have the correct answer. This process not only confirms your solution but also deepens your understanding of the interconnectedness of differentiation and integration.