Integral Formula Verification - Are These Integrals Correct?

Hey guys! Today, we're diving into some integral calculus to figure out if a few formulas are correct. We'll break down each one step-by-step, so you can easily follow along and understand the reasoning. Let's jump right in!

a. ∫(7x + 2)² dx = (7x + 2)³/3 + C

Integrals involving polynomials can be tricky sometimes, so let's analyze this one closely. To determine if this formula is correct, we need to differentiate the result, (7x + 2)³/3 + C, and see if we get back the original integrand, (7x + 2)². If the derivative matches the integrand, then the integral is correct. If not, we know there's a mistake somewhere.

Let's differentiate (7x + 2)³/3 + C using the chain rule. Remember, the chain rule states that if we have a composite function, like f(g(x)), its derivative is f'(g(x)) * g'(x). In our case, the outer function is something cubed (let's call it u³), and the inner function is (7x + 2).

So, differentiating (7x + 2)³/3 + C, we get:

• d/dx [(7x + 2)³/3 + C] = (1/3) * d/dx [(7x + 2)³] + d/dx [C]
• = (1/3) * 3(7x + 2)² * d/dx (7x + 2) + 0 (The derivative of a constant C is always 0)
• = (7x + 2)² * 7
• = 7(7x + 2)²

Now, let's compare this result, 7(7x + 2)², with the original integrand, (7x + 2)². We can clearly see that they are not the same. Our derivative has an extra factor of 7. Therefore, the formula ∫(7x + 2)² dx = (7x + 2)³/3 + C is incorrect.

So, what went wrong? The constant factor is the culprit here. When we integrate, we need to account for the derivative of the inner function (7x + 2), which is 7. The correct integral should include a division by this factor. The constant of integration, C, is also essential because the derivative of a constant is zero, meaning there are infinitely many antiderivatives that differ by a constant.

To correct this, we need to divide the result by 7. So, the correct integral should look something like this: ∫(7x + 2)² dx = (1/7) * (7x + 2)³/3 + C = (7x + 2)³/21 + C. This way, when we differentiate, the factor of 7 from the chain rule will cancel out with the 1/7, giving us the original integrand.

In conclusion, the initial formula missed accounting for the chain rule during integration. Remember, guys, always double-check your work by differentiating the result to ensure it matches the original integrand. It's a crucial step to avoid common integration mistakes!

b. ∫3(7x + 2)² dx = (7x + 2)³ + C

Now, let's tackle the second integral: ∫3(7x + 2)² dx = (7x + 2)³ + C. Again, the key to verifying this formula is differentiation. We'll differentiate the right-hand side, (7x + 2)³ + C, and see if we end up with the integrand, 3(7x + 2)². If they match, the formula is correct; if not, we need to identify the error.

Differentiating (7x + 2)³ + C with respect to x, we use the chain rule again. This is a fundamental technique in calculus, so mastering it is crucial for handling composite functions. The outer function is cubing (u³), and the inner function is (7x + 2).

So, the differentiation process looks like this:

• d/dx [(7x + 2)³ + C] = d/dx [(7x + 2)³] + d/dx [C]
• = 3(7x + 2)² * d/dx (7x + 2) + 0 (Remember, the derivative of a constant is zero)
• = 3(7x + 2)² * 7
• = 21(7x + 2)²

Now, let's compare our result, 21(7x + 2)², with the original integrand, 3(7x + 2)². This time, we see a significant difference. The derivative we calculated has a factor of 21, while the integrand has a factor of only 3. Therefore, the formula ∫3(7x + 2)² dx = (7x + 2)³ + C is incorrect.

The problem here lies in the coefficient. The integral is missing a crucial adjustment to account for the chain rule. When we differentiate (7x + 2)³, we get a factor of 7 from the derivative of the inner function, (7x + 2). To counteract this during integration, we need to divide by 7.

To correct the formula, we need to adjust the result to compensate for this extra factor of 7. The correct integral should look like this:

∫3(7x + 2)² dx = (3/7)(7x + 2)³ + C

If we differentiate (3/7)(7x + 2)³ + C, we get:

• d/dx [(3/7)(7x + 2)³ + C] = (3/7) * 3(7x + 2)² * 7 + 0
• = 3(7x + 2)²

This matches the original integrand, 3(7x + 2)², confirming that our corrected formula is accurate. Remember, guys, always be mindful of the coefficients and how they interact with the chain rule during both differentiation and integration. Small errors can lead to incorrect results.

In summary, the initial formula was incorrect due to the missing coefficient adjustment. By carefully applying the chain rule and verifying the result through differentiation, we were able to pinpoint and correct the error. Let's move on to the final integral now!

c. ∫21(7x + 2)² dx = (7x + 2)³ + C

Finally, let's analyze the third integral: ∫21(7x + 2)² dx = (7x + 2)³ + C. We'll use the same method as before: differentiate the result, (7x + 2)³ + C, and compare it with the integrand, 21(7x + 2)². If they match, the formula is correct; if not, we'll need to identify the error and explain why it occurred.

We've already differentiated (7x + 2)³ + C in the previous example, but let's quickly recap the process to reinforce our understanding. Using the chain rule, we have:

• d/dx [(7x + 2)³ + C] = d/dx [(7x + 2)³] + d/dx [C]
• = 3(7x + 2)² * d/dx (7x + 2) + 0
• = 3(7x + 2)² * 7
• = 21(7x + 2)²

Now, let's compare this result, 21(7x + 2)², with the original integrand, 21(7x + 2)². Guess what? They match perfectly! Therefore, the formula ∫21(7x + 2)² dx = (7x + 2)³ + C is correct.

In this case, the formula is correct because it properly accounts for the chain rule. The constant 21 in the integrand is exactly what we get when we differentiate (7x + 2)³ (3 * 7 = 21). The constant of integration, C, also plays its crucial role, representing the family of antiderivatives.

This example highlights the importance of carefully considering the chain rule when dealing with composite functions in integration. By multiplying the derivative of the outer function by the derivative of the inner function, we ensure that our result, when differentiated, returns the original integrand. There were no missing coefficients or adjustments needed in this formula, making it a straightforward application of the integration rule.

So, guys, this final integral serves as a great example of a correct application of integral calculus. It reinforces our understanding of the chain rule and the role of the constant of integration.

Final Thoughts and Key Takeaways

Alright, guys, we've dissected three integral formulas today, and we've learned some valuable lessons along the way. Let's recap the key takeaways to solidify our understanding:

1. Always Verify by Differentiating: The most reliable way to check if an integral is correct is to differentiate the result. If the derivative matches the original integrand, the integral is correct. This is a fundamental principle of calculus and should be a go-to strategy for every problem.
2. Master the Chain Rule: The chain rule is crucial for differentiating composite functions and is equally important in integration. When dealing with functions like (7x + 2)², remember to account for the derivative of the inner function. This often involves multiplying or dividing by a constant factor.
3. Pay Attention to Coefficients: Constants and coefficients can make or break an integral. Ensure that you're correctly adjusting for them, especially when dealing with the chain rule. A small mistake with a coefficient can lead to a completely wrong answer.
4. Don't Forget the Constant of Integration: The constant of integration, C, is essential because the derivative of a constant is zero. This means there are infinitely many antiderivatives that differ by a constant. Always include + C in your indefinite integrals.

By understanding these key principles, you'll be well-equipped to tackle a wide range of integral problems. Remember, practice makes perfect, so keep working through examples and applying these concepts. You've got this, guys!

If you have any more questions or topics you'd like us to cover, let us know in the comments below. Happy integrating!