Solve Integral (8 + X^8)^8 * X^7 Dx: Step-by-Step Guide

Hey guys! Today, we're diving deep into the fascinating world of calculus to tackle a juicy integral problem. We'll break it down step-by-step, making sure you not only understand the solution but also the underlying concepts. So, buckle up and let's get started!

The Challenge: $\int(8 + x^8)^8 x^7 dx$

Our mission, should we choose to accept it (and we do!), is to evaluate the integral $\int(8 + x^8)^8 x^7 dx$. This might look a bit intimidating at first glance, but don't worry, we've got the tools and techniques to conquer it. The key here is recognizing the structure of the integrand and choosing the right method to simplify it. In this case, we're going to leverage the power of u-substitution, a technique that often works wonders when we have a composite function multiplied by (or related to) its derivative. This method is a cornerstone of integral calculus, allowing us to transform complex integrals into simpler, more manageable forms. Think of it as a mathematical sleight of hand, turning a seemingly impossible problem into a straightforward one.

The Strategy: U-Substitution to the Rescue

The heart of the u-substitution method lies in identifying a suitable 'u' within the integral. The goal is to find a function whose derivative also appears (or can be easily massaged to appear) in the integral. This allows us to rewrite the integral in terms of 'u' and 'du', often resulting in a much simpler expression. For this particular integral, a promising candidate for 'u' is the expression inside the parentheses: $u = 8 + x^8$. Why this choice? Because its derivative, $du$, will involve $x^7$, which we conveniently have in our integrand. Let's see how this unfolds.

Step-by-Step Solution

1. Define u: Let $u = 8 + x^8$. This is our pivotal first step, setting the stage for the rest of the solution. By choosing this particular 'u', we're aiming to simplify the complex expression $(8 + x^8)^8$ and make the integral more approachable.
2. Find du: Now, we differentiate 'u' with respect to 'x': $\frac{du}{dx} = 8x^7$. This gives us a crucial relationship between 'du' and 'dx', which we'll use to rewrite the integral. The appearance of $x^7$ here is no coincidence; it's what makes the u-substitution method so effective in this case.
3. Rewrite dx: We need to express 'dx' in terms of 'du'. From the previous step, we have $du = 8x^7 dx$. Solving for 'dx', we get $dx = \frac{du}{8x^7}$. This is a key step in the substitution process, allowing us to replace 'dx' in the original integral with an expression involving 'du'.
4. Substitute: Now comes the magic! We substitute 'u' and 'dx' into the original integral:

   $$\int (8 + x^8)^8 x^7 dx = \int u^8 x^7 \frac{du}{8x^7}$$

   Notice how the $x^7$ terms cancel out beautifully, leaving us with a much simpler integral in terms of 'u'. This is the power of u-substitution in action.
5. Simplify: After canceling the $x^7$ terms and factoring out the constant, we have:

   $$\int u^8 x^7 \frac{du}{8x^7} = \frac{1}{8} \int u^8 du$$

   This integral looks much more manageable, doesn't it? We've successfully transformed a complex integral into a basic power rule integration.
6. Integrate: Now, we apply the power rule for integration: $\int u^n du = \frac{u^{n+1}}{n+1} + C$. In our case, $n = 8$, so:

   $$\frac{1}{8} \int u^8 du = \frac{1}{8} \cdot \frac{u^9}{9} + C = \frac{u^9}{72} + C$$

   This step is a direct application of a fundamental integration rule, bringing us closer to our final answer.
7. Substitute Back: Finally, we substitute back $u = 8 + x^8$ to express the result in terms of 'x':

   $$\frac{u^9}{72} + C = \frac{(8 + x^8)^9}{72} + C$$

   And there we have it! We've successfully evaluated the integral.

Therefore, $\int (8 + x^8)^8 x^7 dx = \frac{(8 + x^8)^9}{72} + C$, where 'C' is the constant of integration.

Checking Our Work: Differentiation to the Rescue

In calculus, it's always a good idea to check your work, especially with integration. The best way to verify an indefinite integral is to differentiate the result and see if you get back the original integrand. Let's do that now.

1. Differentiate the Result: We need to differentiate $\frac{(8 + x^8)^9}{72} + C$ with respect to 'x'. Let's break it down:

   • The constant 'C' differentiates to 0.
   • For the term $\frac{(8 + x^8)^9}{72}$, we'll use the chain rule. The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

2. Apply the Chain Rule:

   • Let $f(u) = \frac{u^9}{72}$ and $g(x) = 8 + x^8$.
   • Then $f'(u) = \frac{9u^8}{72} = \frac{u^8}{8}$ and $g'(x) = 8x^7$.
   • Applying the chain rule: $\frac{d}{dx} \left[ \frac{(8 + x^8)^9}{72} \right] = f'(g(x)) \cdot g'(x) = \frac{(8 + x^8)^8}{8} \cdot 8x^7$

3. Simplify: The 8s cancel out, leaving us with $(8 + x^8)^8 x^7$.

4. Compare: This is exactly the original integrand! We've successfully verified our result. Woohoo!

Key Takeaways and Insights

• U-Substitution is Your Friend: This technique is a powerful tool for simplifying integrals, especially when you spot a composite function and its derivative (or a multiple thereof) within the integrand. Mastering u-substitution is crucial for tackling a wide range of integration problems.
• Choosing the Right 'u' is Crucial: The success of u-substitution hinges on selecting the appropriate 'u'. Look for functions whose derivatives are present (or can be manipulated to be present) in the integral. Practice and experience will help you develop an intuition for making the right choice.
• Don't Forget the Constant of Integration: Indefinite integrals always have a constant of integration, 'C'. It represents the family of functions that have the same derivative. Forgetting 'C' is a common mistake, so always remember to include it.
• Verification is Key: Always check your work, especially with integration. Differentiating your result is a reliable way to ensure you've found the correct antiderivative. This step can save you from making errors and build confidence in your solutions.
• Practice Makes Perfect: Like any mathematical skill, mastering integration requires practice. Work through various examples, try different techniques, and don't be afraid to make mistakes. Each problem you solve will strengthen your understanding and improve your problem-solving abilities.

Common Pitfalls and How to Avoid Them

• Incorrectly Finding du: Make sure you differentiate 'u' correctly. A mistake in finding 'du' will throw off the entire solution.
• Forgetting to Substitute Back: Remember to substitute back to express your final answer in terms of the original variable ('x' in this case). Leaving the answer in terms of 'u' is an incomplete solution.
• Ignoring the Constant of Integration: Always include the constant of integration, 'C', for indefinite integrals.
• Skipping the Verification Step: It's tempting to skip checking your work, but verification is crucial to catch errors. Take the time to differentiate your result and compare it to the original integrand.
• Giving Up Too Soon: Some integrals are more challenging than others. Don't get discouraged if you don't see the solution immediately. Try different approaches, review the techniques, and persevere. The satisfaction of solving a tough integral is well worth the effort.

Let's Recap

In this article, we've thoroughly explored the evaluation of the integral $\int (8 + x^8)^8 x^7 dx$. We employed the powerful technique of u-substitution, carefully selecting $u = 8 + x^8$ to simplify the integral. We walked through each step, from finding 'du' and rewriting 'dx' to substituting back and expressing the final answer in terms of 'x'. We also emphasized the importance of checking our work by differentiating the result and comparing it to the original integrand. By understanding the underlying concepts, practicing diligently, and avoiding common pitfalls, you'll be well-equipped to tackle a wide range of integration challenges. Keep practicing, keep exploring, and keep those calculus skills sharp!

Practice Problems

To solidify your understanding, try these practice problems:

1. $\int x \sqrt{x^2 + 1} dx$
2. $\int \frac{x^2}{(x^3 + 5)^2} dx$
3. $\int \sin(x) \cos^3(x) dx$

Good luck, and happy integrating!

Conclusion

Evaluating integrals can seem daunting, but with the right techniques and a systematic approach, you can conquer even the most challenging problems. U-substitution is a versatile tool that can simplify many integrals, and understanding its nuances is crucial for success in calculus. Remember to practice, check your work, and never give up. Keep exploring the world of calculus, and you'll be amazed at what you can achieve.