Antiderivatives Of 10cos(x) - 4x: Step-by-Step Solution

Let's dive into the world of antiderivatives! If you're scratching your head trying to figure out how to find the antiderivative of a function like f(x) = 10cos(x) - 4x, you've come to the right place. We'll break it down step by step, making it super easy to understand.

Understanding Antiderivatives

Before we jump into the problem, let's make sure we're all on the same page about what an antiderivative actually is. Think of it as the reverse process of differentiation. If you have a function, its antiderivative is another function whose derivative is the original function. Essentially, we are trying to find a function, let’s call it F(x), such that F'(x) = f(x).

The concept of antiderivatives is a cornerstone of integral calculus. In simpler terms, while derivatives help us understand rates of change, antiderivatives help us find the original function before the change occurred. This has vast applications in physics, engineering, economics, and many other fields. For instance, if you know the velocity of an object, finding the antiderivative gives you the position function. Or, if you have a marginal cost function in economics, the antiderivative will give you the total cost function.

The real magic of antiderivatives lies in the fact that the antiderivative of a function is not unique. Remember that the derivative of a constant is zero? This means that if F(x) is an antiderivative of f(x), then F(x) + C, where C is any constant, is also an antiderivative of f(x). This is why we always add the constant of integration, "C", when finding antiderivatives. The constant "C" represents an entire family of functions that differ only by a constant term, all of which have the same derivative. This constant is crucial because it acknowledges the infinite possible vertical shifts of the antiderivative function.

Key Concepts to Remember

• An antiderivative is the reverse process of differentiation.
• If F(x) is an antiderivative of f(x), then F'(x) = f(x).
• The antiderivative is not unique; we always add a constant of integration, "C".

Now that we've covered the basics, let's get our hands dirty and find the antiderivative of our function.

Breaking Down the Function: f(x) = 10cos(x) - 4x

The function we're working with is f(x) = 10cos(x) - 4x. To find its antiderivative, we'll tackle each term separately. This is a common strategy because the antiderivative of a sum or difference is simply the sum or difference of the antiderivatives.

Breaking down a complex function into smaller, more manageable parts is a crucial problem-solving technique in calculus. It allows us to apply antiderivative rules to individual terms rather than trying to find the antiderivative of the entire expression at once. This approach leverages the linearity property of integrals, which states that the integral of a sum (or difference) is the sum (or difference) of the integrals. In our case, we have two distinct terms: 10cos(x) and -4x. By addressing each term individually, we can apply known antiderivative formulas and then combine the results.

Let's consider the first term, 10cos(x). Here, we have a constant multiple of a trigonometric function. The key to finding its antiderivative lies in remembering the derivatives of trigonometric functions. Specifically, we know that the derivative of sin(x) is cos(x). Therefore, the antiderivative of cos(x) is sin(x). The constant multiple 10 simply carries over, so the antiderivative of 10cos(x) will involve 10sin(x). However, we must also remember to add the constant of integration, C, because there are infinitely many antiderivatives that differ by a constant.

Moving on to the second term, -4x, we encounter a power function. Finding the antiderivative of power functions involves the power rule for integration, which is the reverse of the power rule for differentiation. The power rule for integration states that the antiderivative of x^n is (x^(n+1))/(n+1), provided that n is not equal to -1. In our case, we have -4x, which can be written as -4x^1. Applying the power rule, we increase the exponent by one (1 + 1 = 2) and divide by the new exponent (2). This gives us -4 * (x^2)/2, which simplifies to -2x^2. Again, we must remember to add the constant of integration, C.

By breaking down the function and understanding the individual terms, we've simplified the problem and made it much easier to find the antiderivative. This approach highlights the power of decomposing complex problems into smaller, more manageable parts, a strategy that is applicable in many areas of mathematics and problem-solving.

Finding the Antiderivative of 10cos(x)

Okay, let's focus on the first part: finding the antiderivative of 10cos(x). Remember your basic calculus rules? The antiderivative of cos(x) is sin(x). Since we have a constant multiple of 10, we simply carry that along. So, the antiderivative of 10cos(x) is 10sin(x). But don't forget the constant of integration, C! This gives us 10sin(x) + C.

To truly understand why the antiderivative of cos(x) is sin(x), it's essential to revisit the fundamental relationship between derivatives and antiderivatives. The antiderivative is the reverse operation of differentiation. Therefore, to find the antiderivative of a function, we ask ourselves: "What function, when differentiated, gives us the original function?"

In the case of cos(x), we recall the derivative rules for trigonometric functions. One of the most fundamental rules is that the derivative of sin(x) is cos(x). This direct relationship is the key to finding the antiderivative. Because the derivative of sin(x) is cos(x), it logically follows that the antiderivative of cos(x) is sin(x). This is a cornerstone of calculus and should be firmly planted in your memory.

Now, let's consider the constant multiple of 10 in our term 10cos(x). When dealing with constants in antiderivatives, we rely on the constant multiple rule for integration. This rule states that if you have a constant multiplied by a function, the antiderivative is simply the constant multiplied by the antiderivative of the function. Mathematically, this is expressed as ∫kf(x) dx = k∫f(x) dx, where k is a constant. In our example, the constant is 10, and the function is cos(x). We already know the antiderivative of cos(x) is sin(x). Therefore, we just multiply the constant 10 by sin(x), resulting in 10sin(x).

However, we're not quite finished yet. The most crucial aspect of finding antiderivatives is the constant of integration, denoted by C. This constant arises because the derivative of any constant is zero. This means that when we find an antiderivative, there could have been a constant term in the original function that disappeared during differentiation. For example, the derivative of sin(x) + 5 is also cos(x). The same applies to sin(x) - 100 or sin(x) + π. In each case, the derivative is cos(x), regardless of the constant term. To account for all these possibilities, we add the constant of integration C to the antiderivative. Therefore, the complete antiderivative of 10cos(x) is 10sin(x) + C.

By understanding the relationship between derivatives and antiderivatives, the constant multiple rule, and the importance of the constant of integration, we can confidently find the antiderivative of trigonometric functions like 10cos(x). This foundational knowledge is essential for tackling more complex integration problems in calculus.

Finding the Antiderivative of -4x

Next up, let's tackle the antiderivative of -4x. This involves the power rule for integration. Remember, the power rule states that the antiderivative of x^n is (x^(n+1))/(n+1). In our case, n = 1, so we add 1 to the exponent, giving us 2, and then divide by the new exponent. So, the antiderivative of x is (x^2)/2. Now, we apply this to -4x. We have -4 multiplied by (x^2)/2, which simplifies to -2x^2. And yes, we need to add our constant of integration, C, making it -2x^2 + C.

The power rule for integration is one of the most fundamental and frequently used techniques in calculus. It allows us to find the antiderivative of power functions, which are functions of the form x^n, where n is any real number except -1. Understanding and applying this rule correctly is crucial for mastering integration.

The power rule is essentially the reverse of the power rule for differentiation. The power rule for differentiation states that the derivative of x^n is nx^(n-1). In contrast, the power rule for integration tells us that the antiderivative of x^n is (x^(n+1))/(n+1), provided that n is not equal to -1. The condition n ≠ -1 is important because when n = -1, we have the function 1/x, and its antiderivative is ln|x|, which is a different rule altogether.

To apply the power rule to our term -4x, we first recognize that -4x can be written as -4x^1. Here, the exponent n is 1. The constant -4 is a constant multiple, and we can deal with it separately using the constant multiple rule for integration. Now, we focus on finding the antiderivative of x^1. According to the power rule, we increase the exponent by 1, so 1 + 1 = 2, and then we divide by the new exponent, which is 2. This gives us (x^2)/2.

Next, we bring back the constant multiple -4. We multiply -4 by (x^2)/2, which simplifies to -2x^2. This is the antiderivative of -4x without considering the constant of integration. Remember, the constant multiple rule for integration allows us to pull out constants from the integral, find the antiderivative of the function, and then multiply the result by the constant. This significantly simplifies the process of finding antiderivatives of functions with constant multiples.

Finally, and crucially, we add the constant of integration, C. As discussed earlier, this constant accounts for the fact that the derivative of any constant is zero. Therefore, when we reverse the process and find the antiderivative, we must include C to represent all possible constant terms that could have been present in the original function. Thus, the complete antiderivative of -4x is -2x^2 + C.

The power rule for integration, combined with the constant multiple rule and the constant of integration, provides a powerful tool for finding antiderivatives of a wide range of power functions. Mastering this technique is essential for success in integral calculus and its applications.

Combining the Results

Now we have the antiderivatives of each part: 10sin(x) + C and -2x^2 + C. To find the antiderivative of the entire function f(x) = 10cos(x) - 4x, we simply add these results together. So, the antiderivative is 10sin(x) - 2x^2 + C. Note that we only write one C because the sum of two arbitrary constants is still an arbitrary constant.

When combining the antiderivatives of individual terms, it's crucial to understand why we only need to include a single constant of integration. Each term, when integrated, introduces its own constant of integration. However, since these constants are arbitrary and represent any real number, we can combine them into a single constant. This simplifies the expression and avoids unnecessary complexity.

Let's say we found the antiderivative of 10cos(x) to be 10sin(x) + C1 and the antiderivative of -4x to be -2x^2 + C2, where C1 and C2 are the constants of integration for each term. When we add these antiderivatives together, we get:

10sin(x) + C1 - 2x^2 + C2

We can rearrange this expression as:

10sin(x) - 2x^2 + (C1 + C2)

Since C1 and C2 are both arbitrary constants, their sum (C1 + C2) is also an arbitrary constant. We can represent this new arbitrary constant with a single letter, C. Thus, the combined antiderivative becomes:

10sin(x) - 2x^2 + C

This simplification is valid because the constant of integration represents an entire family of functions that differ only by a constant term. Adding two such families together results in another family of functions that also differ by a constant term. Therefore, using a single constant of integration is sufficient to represent all possible antiderivatives.

Another way to think about this is to consider that if we were to take the derivative of 10sin(x) - 2x^2 + C, the derivative of the constant C would be zero. It doesn't matter whether C represents the sum of two constants or a single constant; the result is the same. This illustrates why we can combine all the constants of integration into one constant without loss of generality.

In summary, when finding the antiderivative of a sum or difference of terms, we find the antiderivative of each term individually, including a constant of integration for each. However, when we combine the results, we only need to write a single constant of integration because the sum of arbitrary constants is itself an arbitrary constant. This simplifies the final expression and accurately represents the family of antiderivatives.

The Final Answer

So, the antiderivative of f(x) = 10cos(x) - 4x is F(x) = 10sin(x) - 2x^2 + C. That's it! You've successfully found the antiderivative of the given function.

Key Takeaways

• Finding antiderivatives involves reversing the process of differentiation.
• The antiderivative of cos(x) is sin(x).
• The power rule for integration is crucial for terms like -4x.
• Always remember the constant of integration, C.

By following these steps and understanding the basic rules, you can tackle many antiderivative problems. Keep practicing, and you'll become a pro in no time!

For further learning and practice problems, you can explore resources like Khan Academy's Integral Calculus section. Happy integrating!