Solving ∫ (3e^(3√y))/(2√y)) Dy: A Step-by-Step Guide

by Andrew McMorgan 53 views

Hey guys! Today, we're diving into a fascinating integral problem that might seem a bit daunting at first glance: ∫ (3e^(3√y))/(2√y)) dy. Don't worry, we'll break it down step by step and make sure you understand every part of the process. So, grab your favorite beverage, get comfy, and let's get started!

Understanding the Integral

Before we jump into solving this integral, let's take a moment to understand what we're dealing with. The integral ∫ (3e^(3√y))/(2√y)) dy involves an exponential function (e^(3√y)) and a square root (√y) in the denominator. This suggests that a u-substitution might be a good approach. Our main goal here is to simplify the integral into a form that we can easily recognize and integrate using standard techniques. Integrals like this often appear in calculus, and mastering them can really boost your math skills. Think of this integral as a puzzle – we just need to find the right pieces and put them together.

Why U-Substitution?

The u-substitution method is a powerful technique for simplifying integrals, especially when we have a composite function (a function within a function). In our case, we have e^(3√y), which is a composite function. The u-substitution allows us to replace a part of the integrand (the function inside the integral) with a new variable, 'u', making the integral simpler to solve. The key is to choose 'u' wisely, so its derivative also appears in the integral, allowing us to cancel out terms. For this integral, selecting u = 3√y seems promising because its derivative will involve 1/√y, which is already present in the integral.

Step-by-Step Solution

Now, let's get our hands dirty and solve this integral step by step. I'll walk you through each stage, making sure you understand the reasoning behind each move. No magical leaps here – just clear, logical steps.

1. Identifying the Right U-Substitution

As we discussed, the first step is to choose an appropriate 'u' for our substitution. Looking at the integral, the term 3√y inside the exponential function seems like a good candidate. So, let's set:

u = 3√y

This choice will help us simplify the exponential part of the integral. Remember, the right u-substitution is the cornerstone of solving these types of problems.

2. Finding du/dy and Solving for dy

Next, we need to find the derivative of 'u' with respect to 'y' (du/dy). This will help us express 'dy' in terms of 'du', which is crucial for the substitution.

u = 3√y = 3y^(1/2)

Now, differentiate 'u' with respect to 'y':

du/dy = 3 * (1/2) * y^((1/2) - 1) = (3/2) * y^(-1/2) = 3 / (2√y)

So, we have:

du/dy = 3 / (2√y)

Now, we need to solve for 'dy'. Multiply both sides by 'dy' and by (2√y)/3 to isolate 'dy':

dy = (2√y / 3) du

This expression for 'dy' is what we'll use to substitute 'dy' in the original integral. It might seem a bit complex now, but you'll see how it simplifies things shortly.

3. Substituting u and du into the Integral

Now comes the exciting part – substituting 'u' and 'dy' into our original integral. Let's rewrite the integral with our substitutions:

∫ (3e^(3√y))/(2√y) dy = ∫ (3e^u) / (2√y) * (2√y / 3) du

Notice how the (2√y) terms in the numerator and denominator cancel each other out, and the 3 in the numerator also cancels out:

∫ (3e^u) / (2√y) * (2√y / 3) du = ∫ e^u du

Wow! Look how much simpler the integral has become. This is the magic of u-substitution at work. We've transformed a complex integral into a basic one that we can easily solve.

4. Integrating with Respect to u

Now that we have the integral in terms of 'u', we can integrate it. The integral of e^u with respect to 'u' is simply e^u:

∫ e^u du = e^u + C

Don't forget the constant of integration, 'C'! It's crucial to include 'C' when evaluating indefinite integrals, as it represents the family of functions that have the same derivative.

5. Substituting Back for y

We're almost there! The last step is to substitute back for 'y'. Remember, we initially set u = 3√y. So, let's replace 'u' with 3√y in our result:

e^u + C = e^(3√y) + C

And that's it! We've successfully evaluated the integral.

Final Answer

The final answer to our integral ∫ (3e^(3√y))/(2√y)) dy is:

e^(3√y) + C

Key Takeaways

Let's recap the key steps we took to solve this integral. This will help solidify your understanding and give you a framework for tackling similar problems in the future.

  1. Identify the Right U-Substitution: Choose a 'u' that simplifies the integral, often a function within a function. In this case, u = 3√y was the key.
  2. Find du/dy and Solve for dy: Calculate the derivative of 'u' with respect to 'y' and solve for 'dy'. This step is crucial for changing the variable of integration.
  3. Substitute u and du into the Integral: Replace parts of the original integral with 'u' and 'du'. This should simplify the integral significantly.
  4. Integrate with Respect to u: Evaluate the simplified integral with respect to 'u'.
  5. Substitute Back for y: Replace 'u' with its original expression in terms of 'y' to get the final answer.

Common Mistakes to Avoid

It's always helpful to be aware of common pitfalls so you can steer clear of them. Here are a few mistakes students often make when dealing with integrals like this:

  • Forgetting the Constant of Integration: Always include '+ C' when evaluating indefinite integrals.
  • Incorrectly Calculating du/dy: Make sure you differentiate 'u' with respect to 'y' accurately. A small mistake here can throw off the entire solution.
  • Not Substituting Back: Don't forget to replace 'u' with its original expression in terms of 'y' at the end. Leaving the answer in terms of 'u' is incomplete.
  • Choosing the Wrong U-Substitution: Selecting the wrong 'u' can make the integral even more complicated. Practice helps in recognizing good candidates for 'u'.

Practice Problems

Practice makes perfect! To really master u-substitution, it's essential to work through a variety of problems. Here are a few practice problems similar to the one we just solved:

  1. ∫ (2x * e(x2)) dx
  2. ∫ (cos(√x) / √x) dx
  3. ∫ (x / √(x^2 + 1)) dx

Try solving these on your own, and feel free to ask if you get stuck. The more you practice, the more comfortable you'll become with these techniques.

Conclusion

So there you have it, guys! We've successfully evaluated the integral ∫ (3e^(3√y))/(2√y)) dy using the method of u-substitution. We broke down each step, highlighted key takeaways, and even discussed common mistakes to avoid. Remember, the key to mastering calculus is practice, practice, practice! Keep working at it, and you'll become a pro in no time. If you have any questions or want to dive deeper into other integral problems, let me know. Happy integrating!