Antiderivative Calculation: Finding General Forms For Y=2x^3 And F(x)=sin(2x+3)

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Hey guys! Today, we're diving into the fascinating world of antiderivatives, a core concept in calculus. We'll be tackling two specific problems: finding the general form of the antiderivative for the function y = 2x³ and for f(x) = sin(2x + 3). Think of antiderivatives as the reverse process of differentiation. If differentiation is like taking a car apart, finding the antiderivative is like putting it back together. Let's get started and unlock these mathematical mysteries!

Finding the General Antiderivative of y = 2x³

When we talk about finding the antiderivative of a function, we're essentially asking, "What function, when differentiated, would give us this original function?" In our case, we want to find a function whose derivative is 2x³. Remember that the antiderivative isn't just one function, but a whole family of functions that differ only by a constant. This constant, usually denoted as C, represents the constant of integration. Let's break down the process step-by-step.

The Power Rule in Reverse

The power rule of differentiation states that if we have a term xⁿ, its derivative is nxⁿ⁻¹. To reverse this process for antiderivatives, we'll apply the power rule in reverse. This means we'll increase the exponent by 1 and then divide by the new exponent. So, for x³, we increase the exponent by 1 to get x⁴, and then we divide by the new exponent, 4. Don't forget, this is a crucial step in antiderivative calculation.

Applying the Constant Multiple Rule

We also have a constant multiple of 2 in front of x³. The constant multiple rule for antiderivatives tells us that we can simply multiply the antiderivative of x³ by this constant. This makes our job much easier, as we can focus on the power rule first and then multiply by the constant.

Putting It All Together

Let's combine these rules to find the antiderivative of 2x³. First, we apply the power rule in reverse to x³, which gives us x⁴/4. Then, we multiply by the constant 2, resulting in 2(x⁴/4), which simplifies to x⁴/2. Finally, we add the constant of integration, C, to account for all possible antiderivatives. So, the general form of the antiderivative of y = 2x³ is x⁴/2 + C. Remember, C can be any constant, making this a family of functions.

Why the Constant of Integration?

The constant of integration C is super important because the derivative of a constant is always zero. This means that when we differentiate x⁴/2 + C, we get 2x³ regardless of the value of C. For example, the derivative of x⁴/2 + 5 is 2x³, and so is the derivative of x⁴/2 - 100. That's why we need to include C to represent all possible antiderivatives.

Finding the Antiderivative of f(x) = sin(2x + 3)

Now, let's tackle a slightly different type of function: f(x) = sin(2x + 3). This involves trigonometric functions and the chain rule in reverse. Finding the antiderivative of trigonometric functions often requires remembering the derivatives of common trig functions and then working backward. It's like being a mathematical detective, piecing together clues to solve the mystery! Understanding these principles is key to antiderivative trigonometric functions.

Remembering Trig Derivatives

We know that the derivative of cos(x) is -sin(x). This is a crucial piece of information for finding the antiderivative of sin(x). However, our function is sin(2x + 3), which involves a composition of functions. This means we'll need to consider the chain rule when finding the antiderivative. Let's delve deeper into how the chain rule affects our calculations.

The Chain Rule in Reverse

The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In reverse, this means we need to account for the derivative of the inner function when finding the antiderivative. In our case, the inner function is (2x + 3), and its derivative is 2. This factor of 2 will play a crucial role in our calculations.

Steps to Find the Antiderivative

  1. Recognize the basic antiderivative: We know the antiderivative of sin(x) is -cos(x). So, we start with -cos(2x + 3).
  2. Account for the chain rule: Since the derivative of (2x + 3) is 2, we need to divide by 2 to undo the chain rule effect. This gives us -1/2 cos(2x + 3).
  3. Add the constant of integration: As always, we add C to represent the family of antiderivatives. This ensures we capture all possible functions whose derivative is sin(2x + 3).

The Final Answer

Putting it all together, the general antiderivative of f(x) = sin(2x + 3) is -1/2 cos(2x + 3) + C. See? It's like solving a puzzle, fitting the pieces together using our knowledge of derivatives and the chain rule. Mastering these techniques is crucial for tackling more complex antiderivative problem-solving.

Verifying Our Answers

One of the best ways to ensure we've found the correct antiderivative is to differentiate our answer and see if we get back the original function. This is like checking our work in reverse, making sure everything adds up. Let's quickly verify our results.

Verifying the Antiderivative of 2x³

We found the antiderivative of 2x³ to be x⁴/2 + C. Let's differentiate this: d/dx (x⁴/2 + C) = 2x³ + 0 = 2x³. Bingo! We got back our original function, so we know our antiderivative is correct. This method of verification is a valuable tool in antiderivative verification.

Verifying the Antiderivative of sin(2x + 3)

We found the antiderivative of sin(2x + 3) to be -1/2 cos(2x + 3) + C. Let's differentiate this: d/dx (-1/2 cos(2x + 3) + C) = -1/2 * (-sin(2x + 3)) * 2 + 0 = sin(2x + 3). Awesome! Again, we got back our original function, confirming our answer. This step-by-step process ensures accuracy and builds confidence in our understanding.

Conclusion: Mastering Antiderivatives

So, guys, we've successfully found the general forms of the antiderivatives for both y = 2x³ and f(x) = sin(2x + 3). We've seen how to apply the power rule in reverse, account for constant multiples, and handle trigonometric functions with the chain rule. Remember, the constant of integration C is super important, and verifying our answers by differentiation is a great way to double-check our work. By understanding these concepts and practicing regularly, you'll become antiderivative masters in no time! This knowledge not only helps in academic settings but also has practical applications, making antiderivative applications a valuable skill to possess.