Solve ∫ [6(3 + 4√x)^8 / √x] Dx: Substitution Method

by Andrew McMorgan 52 views

Hey math enthusiasts! Today, we're diving into the fascinating world of integral calculus, specifically tackling the integral ∫ [6(3 + 4√x)^8 / √x] dx. We'll be using the substitution method, a powerful technique that can simplify complex integrals and make them much easier to solve. So, grab your pencils, and let's get started!

Understanding the Substitution Method

Before we jump into the problem, let's quickly recap the substitution method. The substitution method, also known as u-substitution, is a technique used to simplify integrals by replacing a part of the integrand (the function being integrated) with a new variable, often denoted as u. The goal is to transform the integral into a simpler form that we can readily integrate using standard integration rules.

The key idea behind substitution is the chain rule in differentiation. Recall that the chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In integration, we're essentially working backward, trying to identify a composite function within the integrand and undo the chain rule.

To effectively use substitution, we need to:

  1. Identify a suitable substitution: Look for a part of the integrand (often a function within a function) whose derivative also appears in the integral.
  2. Define u and calculate du: Let u be the chosen substitution, and find its derivative, du/dx. Then, rearrange to express du in terms of dx.
  3. Substitute u and du into the integral: Replace the original variables with u and du, transforming the integral into a form involving only u.
  4. Evaluate the integral in terms of u: Find the antiderivative of the new integrand with respect to u.
  5. Substitute back for x: Replace u with its original expression in terms of x to get the final result.

Applying Substitution to Our Integral: ∫ [6(3 + 4√x)^8 / √x] dx

Now, let's apply these steps to our integral: ∫ [6(3 + 4√x)^8 / √x] dx. The integrand looks a bit intimidating at first, but with a clever substitution, we can simplify it significantly.

1. Identifying a Suitable Substitution

Looking at the integrand, we notice the term (3 + 4√x) raised to the power of 8. This suggests that we might benefit from substituting u for the expression inside the parentheses. Why? Because the derivative of √x is 1/(2√x), which appears in the denominator of the integrand. This is a good sign that substitution will work!

2. Defining u and Calculating du

Let's make the substitution:

  • u = 3 + 4√x

Now, we need to find du. First, let's find the derivative of u with respect to x:

  • du/dx = d/dx (3 + 4√x) = 4 * (1/2) * x^(-1/2) = 2/√x

Next, we rearrange this equation to express du in terms of dx:

  • du = (2/√x) dx

3. Substituting u and du into the Integral

We have our u and du, but notice that our integral has a 6 in the numerator, while our du only has a 2. We need to adjust our du to match the integral. We can do this by multiplying both sides of the du equation by 3:

  • 3du = (6/√x) dx

Now we have a term that perfectly matches a part of our integral! Let's substitute u and 3du into the original integral:

∫ [6(3 + 4√x)^8 / √x] dx = ∫ (3 + 4√x)^8 * (6/√x) dx = ∫ u^8 * 3du = 3 ∫ u^8 du

See how much simpler the integral looks now? We've successfully transformed it into an integral involving only u.

4. Evaluating the Integral in Terms of u

Now we can easily integrate with respect to u. Recall the power rule for integration: ∫ x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get:

3 ∫ u^8 du = 3 * (u^9 / 9) + C = (1/3)u^9 + C

5. Substituting Back for x

We've found the antiderivative in terms of u, but we need our answer in terms of x. So, we substitute back our original expression for u:

(1/3)u^9 + C = (1/3)(3 + 4√x)^9 + C

And there you have it! We've successfully solved the integral.

The Final Result

Therefore, the integral ∫ [6(3 + 4√x)^8 / √x] dx is equal to:

(1/3)(3 + 4√x)^9 + C

Where C is the constant of integration.

Key Takeaways and Tips for Mastering Substitution

  • Practice Makes Perfect: The more you practice substitution, the better you'll become at recognizing suitable substitutions.
  • Look for Composite Functions: Pay close attention to functions within functions. These are often good candidates for substitution.
  • Check Your Derivative: Make sure the derivative of your chosen substitution (or a multiple of it) appears in the integral.
  • Don't Forget the Constant of Integration: Always add "+ C" to your final answer when finding indefinite integrals.
  • Verify Your Answer: You can always check your answer by differentiating it. The result should be the original integrand.

Common Mistakes to Avoid

  • Forgetting to Substitute Back: Make sure to replace u with its original expression in terms of x at the end.
  • Incorrectly Calculating du: Double-check your derivative calculation to avoid errors.
  • Ignoring the Constant of Integration: Omitting "+ C" is a common mistake, especially in indefinite integrals.
  • Choosing the Wrong Substitution: If your initial substitution doesn't simplify the integral, try a different one.

Wrapping Up

The substitution method is a powerful tool in integral calculus, allowing us to solve a wide range of integrals that would otherwise be quite challenging. By carefully choosing our substitution and following the steps outlined above, we can transform complex integrals into simpler forms that are much easier to evaluate. Remember, practice is key to mastering this technique, so keep solving those integrals! We hope this step-by-step guide has been helpful. Keep exploring the exciting world of calculus, and we'll see you in the next mathematical adventure!