Calculus Integration Formula Verification Are These Equations Correct

Hey guys! Today, we're diving deep into some calculus problems to figure out if these integration formulas are correct or not. We'll break down each one step-by-step, so you can follow along and understand the why behind the answers. Let's get started!

a. Is $\int(7x+2)^2 dx = \frac{(7x+2)^3}{3} + C$ Correct?

When we're dealing with integrals, the best way to check if a formula is right is to take the derivative of the result and see if we get back to the original function. So, let's find the derivative of $\frac{(7x+2)^3}{3} + C$ with respect to x.

Taking the Derivative

First off, remember the chain rule: if you have a function inside another function, the derivative involves taking the derivative of the outside function, keeping the inside function the same, and then multiplying by the derivative of the inside function.

Applying this to our problem, let's break it down:

• Our "outside" function is something cubed, divided by 3: $\frac{u^3}{3}$, where u is $(7x+2)$.
• The derivative of $\frac{u^3}{3}$ with respect to u is $u^2$.
• Now, replace u with $(7x+2)$, so we have $(7x+2)^2$.
• Next, we need the derivative of the "inside" function, which is $(7x+2)$. The derivative of $(7x+2)$ with respect to x is simply 7.
• Don't forget the constant of integration, C. The derivative of any constant is 0, so that disappears.

Now, multiplying these parts together (from the chain rule), we get:

$$(7x+2)^2 * 7 = 7(7x+2)^2$$

Comparing with the Original Integral

So, the derivative of $\frac{(7x+2)^3}{3} + C$ is $7(7x+2)^2$. But wait a minute! Our original integral was $\int(7x+2)^2 dx$. These aren't the same! We're off by a factor of 7.

The Verdict

Therefore, the formula $\int(7x+2)^2 dx = \frac{(7x+2)^3}{3} + C$ is incorrect. It's super close, but we're missing that crucial factor of 7. To make it right, we'd need to adjust the constant in front of the $(7x+2)^3$ term.

b. Is $\int 3(7x+2)^2 dx = (7x+2)^3 + C$ Correct?

Let's use the same method: take the derivative of the right-hand side and see if we get back to the original integrand. This time, we're checking if the derivative of $(7x+2)^3 + C$ equals $3(7x+2)^2$.

Differentiating the Result

Again, we need the chain rule. Our "outside" function is something cubed, so the derivative will involve bringing down the 3 and reducing the power by 1. The "inside" function is $(7x+2)$.

• The derivative of $u^3$ (where u is $(7x+2)$) with respect to u is $3u^2$.
• Replacing u, we have $3(7x+2)^2$.
• The derivative of the inside, $(7x+2)$, with respect to x is 7.

Putting it all together using the chain rule, we have:

$$3(7x+2)^2 * 7 = 21(7x+2)^2$$

Comparing with the Original Integral

Hmm, the derivative of $(7x+2)^3 + C$ is $21(7x+2)^2$. But our original integral was $\int 3(7x+2)^2 dx$. They don't match! We're off by a factor again, but this time it’s a factor of 7 in the other direction.

The Final Word

So, the formula $\int 3(7x+2)^2 dx = (7x+2)^3 + C$ is also incorrect. It’s another near miss, highlighting the importance of getting those constants right in integration problems.

c. Is $\int 21(7x+2)^2 dx = (7x+2)^3 + C$ Correct?

Okay, third time's the charm, right? Let's see if this formula holds up. We're checking if the derivative of $(7x+2)^3 + C$ is equal to $21(7x+2)^2$.

Taking the Derivative (Again!)

We actually already did this in part (b)! We found that the derivative of $(7x+2)^3 + C$ is indeed $21(7x+2)^2$. This is because:

• The derivative of $(7x+2)^3$ (using the chain rule) is $3(7x+2)^2 * 7 = 21(7x+2)^2$.
• The derivative of the constant C is 0.

Matching the Original Integral

Guess what? The derivative we calculated, $21(7x+2)^2$, perfectly matches the integrand in our original integral, $\int 21(7x+2)^2 dx$!

The Verdict

Finally! The formula $\int 21(7x+2)^2 dx = (7x+2)^3 + C$ is correct. We’ve found a winner!

Key Takeaways for Integration

So, what did we learn from all this? Here are some important things to keep in mind when you're tackling integration problems:

1. Always check your work by differentiating. It's the best way to be sure you've got the right answer. If the derivative of your result doesn't match the original integrand, you've made a mistake.
2. Pay close attention to constants. Integration is tricky because constants can easily get lost or miscalculated. The chain rule, in particular, often introduces constants that you need to account for. As we saw in the examples, a missing or incorrect constant can make the entire formula wrong.
3. Understand the chain rule inside and out. The chain rule is fundamental to differentiating composite functions (functions inside other functions). Since integration is the reverse of differentiation, the chain rule plays a crucial role in integration problems as well. Make sure you're comfortable applying it both ways.
4. Practice, practice, practice! The more you work through integration problems, the better you'll become at recognizing patterns and avoiding common mistakes. Don’t be afraid to make mistakes – they’re part of the learning process. Each time you identify an error, you’re reinforcing the correct method in your mind.

By practicing and paying attention to these key points, you’ll be well on your way to mastering integration!

Final Thoughts

I hope this breakdown helped you guys understand how to check integration formulas and the importance of the chain rule and constants. Keep practicing, and you’ll become calculus pros in no time! Remember, the key is to understand the concepts, not just memorize the formulas. Happy integrating!