Hey guys! Today, we're diving into the fascinating world of calculus to tackle an indefinite integral. Specifically, we're going to figure out how to solve this bad boy: ∫x⁴√(10+x⁵) dx. Don't worry if it looks intimidating at first; we'll break it down step by step, making it super easy to understand. So, grab your calculators (or maybe just your thinking caps!), and let's get started!
Understanding Indefinite Integrals
Before we jump into the nitty-gritty, let's quickly recap what indefinite integrals are all about. An indefinite integral, in essence, is the reverse process of differentiation. Think of it as asking the question, "What function, when differentiated, gives us this expression inside the integral?" The answer isn't just one function, but a whole family of functions that differ only by a constant. That's why we always add the "+ C" at the end, where C represents the constant of integration. This constant acknowledges the fact that the derivative of a constant is zero, so any constant term would disappear during differentiation.
The expression inside the integral, in our case x⁴√(10+x⁵), is called the integrand. Our mission is to find a function whose derivative is equal to this integrand. This process often involves clever techniques and a bit of algebraic manipulation. But don't fret! We'll use a powerful technique called u-substitution to make this integral much more manageable. This method helps us simplify complex integrals by replacing a part of the integrand with a new variable, making the integral easier to solve.
Why is understanding indefinite integrals so important, you ask? Well, they form the foundation for many applications in science and engineering. From calculating areas and volumes to modeling physical phenomena like motion and growth, indefinite integrals are indispensable tools. Mastering these concepts will not only help you ace your calculus exams but also equip you with problem-solving skills applicable across various fields. Now, let's get our hands dirty and see how u-substitution works in practice.
The Magic of U-Substitution
The u-substitution technique, also known as substitution or change of variables, is a powerful tool for simplifying integrals, particularly those involving composite functions. The core idea is to identify a suitable part of the integrand (let's call it 'u') and substitute it, along with its differential 'du', to transform the integral into a simpler form. This often makes the integral look like one we readily know how to solve using basic integration rules.
So, how do we choose the right 'u'? A good strategy is to look for a function within the integrand whose derivative also appears in the integral (possibly with a constant factor). In our integral, ∫x⁴√(10+x⁵) dx, we can see a composite function: √(10+x⁵). The 'inner' function here is 10+x⁵. What's its derivative? Well, the derivative of 10 is 0, and the derivative of x⁵ is 5x⁴. Notice anything familiar? We have x⁴ in our integrand! This is a strong indicator that u-substitution with u = 10+x⁵ might be the key.
Now, let's see how the magic unfolds. If we let u = 10 + x⁵, we need to find du, which is the derivative of u with respect to x, multiplied by dx. So, du = (5x⁴) dx. Looking back at our integral, we have x⁴ dx, but we need 5x⁴ dx to perfectly match our du. No problem! We can manipulate the equation by dividing both sides by 5, giving us (1/5) du = x⁴ dx. Now we have everything we need to substitute! We'll replace 10+x⁵ with 'u' and x⁴ dx with (1/5) du, transforming our integral into a much simpler expression. This transformation is the heart of u-substitution, allowing us to tackle integrals that initially seemed daunting.
Applying U-Substitution to Our Integral
Okay, guys, let's put our u-substitution skills to work on our integral: ∫x⁴√(10+x⁵) dx. Remember, we identified that a good choice for 'u' is the expression inside the square root, u = 10 + x⁵. We also found that du = 5x⁴ dx, and consequently, (1/5) du = x⁴ dx. Now we're ready to transform the integral.
First, we'll replace the (10 + x⁵) inside the square root with our 'u', so √(10+x⁵) becomes √u. Then, we'll replace x⁴ dx with (1/5) du. Our integral now looks like this: ∫√u * (1/5) du. This is already a significant improvement! We can pull the constant (1/5) out of the integral, giving us (1/5) ∫√u du.
Now, let's rewrite the square root of 'u' as u raised to the power of 1/2: (1/5) ∫u^(1/2) du. This is a basic power rule integral! The power rule for integration states that ∫xⁿ dx = (x^(n+1))/(n+1) + C, where n ≠ -1. Applying this rule to our integral, we get:
(1/5) * (u^((1/2)+1)) / ((1/2)+1) + C
Simplifying the exponents, we have (1/5) * (u^(3/2)) / (3/2) + C. To divide by a fraction, we multiply by its reciprocal, so we get (1/5) * (2/3) * u^(3/2) + C. Multiplying the constants, we have (2/15)u^(3/2) + C. We're almost there! Remember, our original integral was in terms of 'x', so we need to substitute back our expression for 'u'. This final step will give us the solution in terms of our original variable.
The Final Substitution and Solution
Alright, team, we've done the heavy lifting! We've successfully transformed our integral using u-substitution and applied the power rule. Now, the final step is to substitute back for 'u' to express our answer in terms of 'x'. We recall that we defined u = 10 + x⁵. So, we simply replace 'u' in our expression (2/15)u^(3/2) + C with (10 + x⁵).
This gives us the final result: (2/15)(10 + x⁵)^(3/2) + C. And there you have it! We've successfully evaluated the indefinite integral of x⁴√(10+x⁵) dx. The answer is (2/15)(10 + x⁵)^(3/2) + C. Remember that '+ C' is crucial because it represents the constant of integration, acknowledging the family of functions that have the same derivative.
Let's take a moment to appreciate what we've accomplished. We started with a seemingly complex integral and, by strategically using u-substitution, we transformed it into a manageable form. We then applied the power rule of integration and substituted back to get our final answer. This process highlights the power and elegance of calculus techniques. If you want to double-check your work (which is always a good idea!), you can differentiate our result and see if it matches the original integrand. Differentiation is the reverse process of integration, so taking the derivative of (2/15)(10 + x⁵)^(3/2) + C should lead us back to x⁴√(10+x⁵). Doing this verification step can provide confidence in your solution.
Checking Our Answer (Bonus!)
Hey guys, just a quick bonus to really solidify our understanding! Let's check if our answer, (2/15)(10 + x⁵)^(3/2) + C, is indeed correct. To do this, we'll differentiate it and see if we get back our original integrand, x⁴√(10+x⁵).
First, we'll differentiate (2/15)(10 + x⁵)^(3/2). We'll use the chain rule, which states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In our case, f(u) = (2/15)u^(3/2) and g(x) = 10 + x⁵.
The derivative of f(u) with respect to u is (2/15) * (3/2) * u^(1/2) = (1/5)u^(1/2). The derivative of g(x) with respect to x is 5x⁴. Applying the chain rule, the derivative of (2/15)(10 + x⁵)^(3/2) is (1/5)(10 + x⁵)^(1/2) * 5x⁴.
Simplifying, we get x⁴(10 + x⁵)^(1/2), which is the same as x⁴√(10 + x⁵). And, of course, the derivative of the constant C is 0. So, the derivative of our result (2/15)(10 + x⁵)^(3/2) + C is indeed x⁴√(10 + x⁵), which is our original integrand! This confirms that our integration was done correctly.
This step is super important because it allows us to verify our solution. Differentiation and integration are inverse operations, so differentiating the result of an indefinite integral should always lead us back to the original function inside the integral. This check not only builds confidence in our answer but also reinforces our understanding of the fundamental relationship between differentiation and integration. Keep up the great work, everyone! You're becoming integral-solving pros!