Evaluating The Integral Of (ln(x))^26 / X A Comprehensive Guide

Hey guys! Today, we're diving into a fun and slightly challenging indefinite integral problem. We're going to evaluate the integral of (ln(x))^26 / x with respect to x. Don't forget, we always need to remember our constant of integration! So, let's break it down step by step.

Understanding the Integral: ∫(ln(x))^26 / x dx

When we first look at the integral ∫(ln(x))^26 / x dx, it might seem a bit intimidating because of the (ln(x))^26 term. But don't worry! We've got a trusty technique up our sleeves called u-substitution that's going to make our lives much easier. The key here is to identify a part of the integrand whose derivative is also present, which will allow us to simplify the integral. In this case, we can see that the derivative of ln(x) is 1/x, and we have that 1/x sitting right there in our integral. This is a classic setup for u-substitution, making it a perfect candidate for this method.

The u-substitution method works by replacing a function within the integral with a new variable, u, and then transforming the integral into a simpler form that we can easily solve. By choosing u = ln(x), we not only simplify the complex (ln(x))^26 term but also naturally incorporate the 1/x term, which will be part of our du. This is the magic of u-substitution: it allows us to untangle complicated integrals by making strategic variable changes. Once we find the integral in terms of u, we simply substitute back to get the result in terms of x. This process makes seemingly difficult integrals manageable and is a powerful tool in our calculus toolkit. So, with our u-substitution strategy in mind, let's move on to the actual substitution and see how this simplifies our problem.

Applying U-Substitution: A Detailed Walkthrough

Okay, let's get our hands dirty with the actual substitution. As we discussed, the smart move here is to let u = ln(x). This choice is crucial because it simplifies the (ln(x))^26 part and prepares us to deal with the 1/x term. Now, the next step in u-substitution is to find du, which is the derivative of u with respect to x, multiplied by dx. So, if u = ln(x), then du/dx = 1/x. To get du, we multiply both sides by dx, giving us du = (1/x) dx. This is fantastic news because we see the (1/x) dx term in our original integral! This means our substitution is perfectly set up to simplify things beautifully.

Now that we have u = ln(x) and du = (1/x) dx, we can rewrite our integral in terms of u. The original integral, ∫(ln(x))^26 / x dx, transforms into ∫u^26 du. Notice how much simpler this looks! The complicated (ln(x))^26 term has become a straightforward u^26, and the 1/x dx part has neatly been replaced by du. This is the power of u-substitution in action. We've taken a complex integral and turned it into a basic power rule problem. The transformation highlights the elegance of this technique, allowing us to tackle integrals that initially appear daunting. With our simplified integral in hand, we are now ready to apply the power rule for integration and find the antiderivative of u^26. Let's move on to that step and see how easily we can solve this new, simplified integral.

Integrating with Respect to U: The Power Rule

Alright, we've transformed our integral into ∫u^26 du, which looks much friendlier, right? Now, it's time to apply the power rule for integration. Remember, the power rule states that ∫x^n dx = (x^(n+1)) / (n+1) + C, where C is the constant of integration. This rule is a cornerstone of integral calculus, and it's going to help us solve this integral in a snap.

Applying the power rule to ∫u^26 du, we simply add 1 to the exponent (26 + 1 = 27) and then divide by the new exponent. This gives us (u^27) / 27. And of course, we can't forget our constant of integration, C, because the derivative of a constant is zero, meaning that there could be any constant term present in the original function. So, the integral of u^26 with respect to u is (u^27) / 27 + C. This step is straightforward, but it's crucial to remember the power rule and the constant of integration to ensure we have the correct antiderivative. Now that we've found the integral in terms of u, our final step is to substitute back for x and get our solution in terms of the original variable. This will complete our u-substitution process and give us the final answer to our integral problem. So, let's proceed with the back-substitution and wrap this up.

Substituting Back to X: Completing the Solution

We're almost there, guys! We've successfully integrated with respect to u, and we have (u^27) / 27 + C. Now, the final step is to substitute back for x to get our answer in terms of the original variable. Remember, we set u = ln(x), so we're going to replace u with ln(x) in our expression. This is a critical step because we need our final answer to be in the same terms as the original problem. Substituting ln(x) for u, we get ((ln(x))^27) / 27 + C. This expression represents the indefinite integral of (ln(x))^26 / x with respect to x.

This final substitution ties everything together, showing the complete solution process from the initial integral to the final antiderivative. It's a testament to the power of u-substitution, which allowed us to transform a seemingly complex integral into a manageable form. The constant of integration, C, remains an essential part of our answer, reminding us that there are infinitely many antiderivatives that differ only by a constant. So, with this final step, we have successfully evaluated the indefinite integral, and we have a complete and accurate solution. Let's recap our steps and present the final answer clearly.

Final Answer: Putting It All Together

Okay, let's recap what we've done and present our final answer. We started with the integral ∫(ln(x))^26 / x dx. We recognized that u-substitution was the perfect technique for this problem. We let u = ln(x), which gave us du = (1/x) dx. This transformation allowed us to rewrite the integral as ∫u^26 du. Applying the power rule, we found that the integral of u^26 is (u^27) / 27 + C. Finally, we substituted back for x, replacing u with ln(x), to get our final answer.

Therefore, the indefinite integral of (ln(x))^26 / x dx is ((ln(x))^27) / 27 + C. Remember, C is the constant of integration. This is our final, comprehensive solution. We've successfully navigated this integral using u-substitution, showcasing a powerful technique in calculus. By breaking the problem down into manageable steps, we were able to solve it methodically and accurately. Great job, guys! I hope this step-by-step guide has been helpful in understanding how to tackle similar integrals. Keep practicing, and you'll become a pro at these in no time!

So, to be crystal clear, here's our final, final answer:

∫(ln(x))^26 / x dx = ((ln(x))^27) / 27 + C

Repair Input Keyword

Evaluate the indefinite integral ∫ (ln(x))^26 / x dx. (Remember to include the constant of integration.)