Solving ∫x²(x³-2)¹³ Dx: A Step-by-Step Guide

by Alex Johnson 45 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of calculus to tackle a specific integral: ∫x²(x³-2)¹³ dx. Don't worry if it looks intimidating at first glance; we're going to break it down step by step using a clever technique called u-substitution. This method is your best friend when dealing with integrals where you spot a composite function and its derivative (or a multiple of it) lurking within the integrand. Ready to unravel this integral? Let's get started!

Understanding the U-Substitution Method

Before we jump into the problem, let's quickly recap the u-substitution method. At its core, u-substitution is the reverse of the chain rule in differentiation. It allows us to simplify complex integrals by replacing a part of the integrand with a new variable, 'u'. The goal is to transform the integral into a simpler form that we can easily integrate using standard rules. The key is to choose your 'u' wisely! Often, we look for a function within the integral whose derivative is also present (up to a constant multiple).

When you are thinking about integrals, the u-substitution method is a technique to simplify integrals where the integrand is a composite function. This method is the reverse of the chain rule in differentiation. The core idea is to replace a part of the integrand with a new variable, u, to make the integral easier to solve. Selecting the right u is crucial; you should look for a function within the integral whose derivative is also present (possibly with a constant multiple). Once you've chosen u, you'll compute du, rewrite the integral in terms of u, solve the simpler integral, and finally, substitute back to express the result in terms of the original variable. This approach simplifies many complex integrals, making them manageable with basic integration rules. So, let's keep this in mind as we move through the steps to solve our integral today.

Step-by-Step Solution: ∫x²(x³-2)¹³ dx

Now, let’s apply this to our integral: ∫x²(x³-2)¹³ dx.

1. Choose Your 'u'

Look closely at the integral. Do you see a function and its derivative (or a multiple thereof)? The expression (x³-2) looks promising because its derivative is 3x², which is a multiple of x² – a term already present in the integral! So, let’s make our substitution:

u = x³ - 2

2. Find 'du'

Next, we need to find the differential 'du'. This simply means taking the derivative of 'u' with respect to 'x' and then multiplying by 'dx':

du/dx = 3x²

du = 3x² dx

3. Adjust the Integral

Notice that our integral has x² dx, but our 'du' is 3x² dx. No problem! We can easily adjust this by dividing both sides of the 'du' equation by 3:

(1/3) du = x² dx

Now we have an exact match for the x² dx term in our original integral.

4. Substitute and Rewrite the Integral

Time to substitute! Replace (x³-2) with 'u' and x² dx with (1/3) du. Our integral now transforms into:

∫(1/3)u¹³ du

See how much simpler that looks? We've successfully transformed the integral into a form that's much easier to handle.

The substitution method is a powerful tool in calculus, especially when you need to evaluate integrals of composite functions. The key is to identify a suitable substitution that simplifies the integral into a more manageable form. Let's revisit our problem: ∫x²(x³-2)¹³ dx. We strategically chose u = x³ - 2 because its derivative, 3x², is present in the integral (up to a constant multiple). This choice is crucial because it allows us to replace both the composite function (x³-2)¹³ and the x² dx term with simpler expressions in terms of u. Once we find du = 3x² dx, we adjust the integral by rewriting x² dx as (1/3) du. This substitution transforms the original integral into ∫(1/3)u¹³ du, which is a much simpler power rule integral. Remember, the goal is to make the integration process easier by transforming the integral into a form that can be solved using standard techniques. Choosing the correct substitution is often the most challenging part, but with practice, you'll become more adept at spotting the right u.

5. Integrate

Now we can integrate with respect to 'u'. Remember the power rule for integration: ∫uⁿ du = (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration.

(1/3) ∫u¹³ du = (1/3) * (u¹⁴/14) + C

Simplify this a bit:

(1/42)u¹⁴ + C

6. Substitute Back

We're not done yet! Our integral is solved in terms of 'u', but we need the answer in terms of our original variable, 'x'. So, we substitute back u = x³ - 2:

(1/42)(x³ - 2)¹⁴ + C

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

When you are performing integration, always remember to substitute back to the original variable after integrating. In our case, we found the integral in terms of u, but the original problem was in terms of x. Substituting back ensures that our final answer is expressed in the same variable as the original integral. So, after integrating ∫(1/3)u¹³ du and getting (1/42)u¹⁴ + C, we replaced u with x³ - 2 to get the final answer: (1/42)(x³ - 2)¹⁴ + C. This step is essential for completing the problem correctly and providing the answer in the context of the original question. Always double-check to make sure you've substituted back correctly to avoid errors. This practice will help you master the u-substitution method and tackle complex integrals with confidence.

Final Answer

Therefore, the integral ∫x²(x³-2)¹³ dx evaluates to (1/42)(x³ - 2)¹⁴ + C, where C is the constant of integration. Congratulations, you've just conquered a challenging integral using u-substitution!

Key Takeaways

Let's recap the key steps to solidify your understanding of the u-substitution method:

  • Identify the composite function: Look for a function within a function.
  • Choose 'u': Select 'u' as the inner function, often the one whose derivative is also present (or a multiple of it).
  • Find 'du': Calculate the derivative of 'u' with respect to 'x' and multiply by 'dx'.
  • Adjust the integral: Manipulate 'du' if necessary to match the remaining terms in the integral.
  • Substitute: Replace the original terms with 'u' and 'du' to rewrite the integral.
  • Integrate: Evaluate the simpler integral in terms of 'u'.
  • Substitute back: Replace 'u' with its original expression in terms of 'x'.
  • Add the constant of integration: Don't forget the '+ C'!

The key to mastering integration using u-substitution lies in recognizing patterns and practicing regularly. The ability to identify the right 'u' is crucial, and this comes with experience. The more integrals you solve, the better you'll become at spotting suitable substitutions. Remember, the goal is to simplify the integral into a form that you can easily integrate using basic rules. So, when you encounter a complex integral, take a moment to analyze the integrand and look for composite functions and their derivatives. With practice, you'll develop an intuition for choosing the right 'u' and transforming challenging integrals into solvable problems. Keep practicing, and you'll become more confident in your integration skills.

Practice Makes Perfect

The best way to master u-substitution is to practice! Try tackling similar integrals, and don't be afraid to experiment with different choices for 'u'. The more you practice, the more intuitive the process will become.

Conclusion

So, there you have it! We've successfully evaluated the integral ∫x²(x³-2)¹³ dx using the u-substitution method. This technique is a powerful tool in your calculus arsenal, allowing you to tackle complex integrals with confidence. Remember the steps, practice regularly, and you'll be well on your way to mastering integration!

For further learning on integration techniques, consider exploring resources like Khan Academy's Calculus lessons. Happy integrating!