Antiderivative Of (e^x + 2e^(-2x)) / (e^x - E^(-2x))^3: Solved!

by Andrew McMorgan 64 views

Hey math enthusiasts! Today, we're diving into an exciting calculus problem: finding the antiderivative of the function (e^x + 2e^(-2x)) / (e^x - e(-2x))3. This might look a bit intimidating at first glance, but don't worry, we'll break it down step by step. So, grab your pencils, and let's get started!

Understanding Antiderivatives

Before we jump into the solution, let's quickly recap what antiderivatives are. In simple terms, an antiderivative is the reverse process of differentiation. If we have a function f(x), its antiderivative, often denoted as F(x), is a function whose derivative is f(x). In other words, F'(x) = f(x). Finding antiderivatives is a fundamental concept in integral calculus, and it's super useful in various applications, from physics to engineering.

The antiderivative is also known as the indefinite integral. It represents a family of functions that differ by a constant. When we find an antiderivative, we usually add a constant of integration, denoted as 'C', to account for this family of functions. This constant is crucial because the derivative of a constant is always zero, meaning there could be an infinite number of constants that would satisfy the antiderivative.

When faced with a complex function like the one we're tackling today, different techniques can be employed to find the antiderivative. One common technique is u-substitution, which we'll use in this case. Another approach is integration by parts, which is particularly useful when dealing with products of functions. The choice of method often depends on the structure of the function and what substitutions might simplify the integral.

Problem Setup: Defining the Derivative

Alright, let's get down to business. Our mission, should we choose to accept it (and we do!), is to find the antiderivative, denoted as p, of the given derivative:

dp/dx = (e^x + 2e^(-2x)) / (e^x - e^(-2x))^3

This looks like a job for u-substitution, a technique that simplifies integration by replacing a complex expression with a single variable. The key here is to identify a suitable 'u' that, when substituted, will make the integral more manageable. In this case, the denominator (e^x - e^(-2x)) seems like a good candidate because its derivative appears in the numerator (with a bit of tweaking).

So, the first step is to carefully examine the function and identify a part of it that, when differentiated, yields another part of the function (up to a constant multiple). This is crucial for setting up the u-substitution correctly. The goal is to transform the integral into a simpler form that can be easily integrated using standard techniques.

It's also helpful to remember the basic derivatives of exponential functions. The derivative of e^x is simply e^x, and the derivative of e^(kx) is k*e^(kx), where k is a constant. These rules are essential when calculating the derivative of the substituted expression and ensuring the substitution simplifies the integral as intended.

The Strategy: U-Substitution

Okay, guys, let's use u-substitution! This is a fantastic technique for simplifying integrals, and it's perfect for this problem. Here’s how we'll approach it:

  1. Identify 'u': We'll let u = e^x - e^(-2x). This choice is strategic because the derivative of u is closely related to the numerator of our fraction.
  2. Calculate du/dx: We need to find the derivative of u with respect to x. This will help us relate du and dx.
  3. Solve for dx: We'll rearrange the equation from step 2 to express dx in terms of du. This is crucial for substituting dx in the original integral.
  4. Substitute: We'll replace (e^x - e^(-2x)) with u and dx with its equivalent expression in terms of du in the original integral. This will transform the integral into a simpler form.
  5. Integrate: We'll integrate the new expression with respect to u. This should be a much easier integral to solve.
  6. Substitute back: Finally, we'll replace u with its original expression in terms of x to get the antiderivative in terms of x.

This step-by-step approach will help us systematically solve the integral. U-substitution is a powerful tool, but it's important to perform each step carefully to avoid errors. The goal is to transform a complex integral into a simpler one that can be easily solved using basic integration rules.

Step-by-Step Solution

Let's roll up our sleeves and get into the nitty-gritty of solving this problem. Here’s the breakdown:

  1. Let u = e^x - e^(-2x)

    We've made our choice for u. Now, let’s move on to the next step.

  2. Calculate du/dx:

    du/dx = d/dx (e^x - e^(-2x))
      = e^x - (-2)e^(-2x)
      = e^x + 2e^(-2x)

    Aha! Notice that du/dx is exactly the numerator of our original fraction. This is a good sign – it means our u-substitution choice is likely to work out nicely.

  3. Solve for dx:

du = (e^x + 2e^(-2x)) dx dx = du / (e^x + 2e^(-2x)) ```

Now we have *dx* expressed in terms of *du*. This is a key step in the substitution process.

  1. Substitute:

    Let's substitute u and dx into our original integral:

    ∫ (e^x + 2e^(-2x)) / (e^x - e^(-2x))^3 dx = ∫ (e^x + 2e^(-2x)) / u^3  [du / (e^x + 2e^(-2x))]

    Notice how the (e^x + 2e^(-2x)) terms cancel out? That’s the magic of u-substitution! We're left with:

    ∫ 1/u^3 du = ∫ u^(-3) du

    This integral looks much simpler, doesn't it?

  2. Integrate:

    Now we can easily integrate with respect to u:

    ∫ u^(-3) du = u^(-2) / (-2) + C
          = -1 / (2u^2) + C

    Remember to add the constant of integration, C! It's a crucial part of finding the general antiderivative.

  3. Substitute back:

    Finally, let's substitute u = e^x - e^(-2x) back into our expression:

    p = -1 / [2(e^x - e^(-2x))^2] + C

    And there you have it! We've found the antiderivative.

The Final Answer

So, the antiderivative p of the given derivative dp/dx = (e^x + 2e^(-2x)) / (e^x - e(-2x))3 is:

p = -1 / [2(e^x - e^(-2x))^2] + C

Where C is the constant of integration. Pat yourself on the back – you've conquered a challenging calculus problem!

Key Takeaways

Let's recap the key takeaways from this problem:

  • U-substitution is your friend: This technique is incredibly powerful for simplifying integrals. Look for expressions and their derivatives within the integrand to guide your choice of u.
  • Careful with the algebra: Make sure to correctly calculate du/dx and solve for dx. A small mistake in the algebra can throw off the entire solution.
  • Don't forget the constant of integration: Always add C when finding an indefinite integral. This accounts for the family of functions that have the same derivative.
  • Practice makes perfect: The more you practice these types of problems, the better you'll become at recognizing patterns and choosing the right techniques.

Wrapping Up

Great job, everyone! We've successfully navigated a tricky antiderivative problem using u-substitution. Remember, calculus can seem daunting, but breaking it down into manageable steps makes it much easier. Keep practicing, keep exploring, and you'll become a calculus whiz in no time! If you have any questions or want to tackle another problem, let me know in the comments below. Keep those mathematical gears turning!

I hope this step-by-step explanation was helpful. Until next time, happy integrating!