Calculate Integral: ∫(2x³ + X)(x⁴ + X²)⁴ Dx

Alright, guys, let's dive into calculating this integral: ∫(2x³ + x)(x⁴ + x²)⁴ dx. It looks a bit intimidating at first, but don't worry, we'll break it down step by step. The key here is recognizing a suitable substitution. Our handwritten note suggests M = x⁴ + x², and that's precisely the path we'll take. This substitution will simplify the integral and make it much easier to solve.

Understanding the Substitution

So, why does the substitution M = x⁴ + x² work? Well, let's think about it. When we have an integral that looks complex, we often look for parts of the expression whose derivatives also appear in the integral. In this case, if M = x⁴ + x², then dM/dx = 4x³ + 2x. Notice anything familiar? That's right! 2x³ + x is almost exactly half of that derivative! This is crucial because it means we can rewrite the integral in terms of M and dM, significantly simplifying the expression. Substitution is a powerful technique in calculus that allows us to transform complicated integrals into more manageable forms. By identifying a suitable substitution, we can often unravel the complexities of the original integral and find a solution more easily. Remember, the goal of substitution is to replace a part of the integral with a new variable (in this case, 'M') such that the derivative of that new variable also appears in the integral, allowing us to simplify the expression and solve it more readily. It's like finding a secret key that unlocks the solution to the puzzle. Now, let's proceed with applying this substitution to our integral and see how it simplifies the calculation process.

Performing the Substitution

Okay, let's get our hands dirty! We have M = x⁴ + x², which means dM/dx = 4x³ + 2x. We want to express our original integral in terms of M and dM. Notice that our integral has (2x³ + x) dx. We can rewrite this as ½ (4x³ + 2x) dx. Now, we can clearly see that (4x³ + 2x) dx is just dM. Therefore, (2x³ + x) dx = ½ dM. Now we can rewrite the entire integral in terms of M:

∫(2x³ + x)(x⁴ + x²)⁴ dx = ∫(M⁴) (½ dM) = ½ ∫M⁴ dM

See how much simpler that looks? The original integral, with its polynomial terms and exponents, has been transformed into a much more straightforward integral involving only M⁴. This transformation is the essence of substitution – to simplify the integral by replacing complex expressions with simpler variables. By expressing the integral in terms of M and dM, we've effectively eliminated the complexity of the original expression and made it much easier to integrate. Now, we can proceed with integrating M⁴ with respect to M, which is a straightforward application of the power rule for integration. This substitution technique is a powerful tool in calculus, allowing us to tackle integrals that would otherwise be difficult or impossible to solve directly. So, with this simplified form, we're well on our way to finding the solution to our integral.

Integrating with Respect to M

Now for the fun part – integration! We have ½ ∫M⁴ dM. Using the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, we can easily integrate M⁴ with respect to M:

½ ∫M⁴ dM = ½ (M⁵/5) + C = M⁵/10 + C

Where C is the constant of integration. Remember, the constant of integration is always added when evaluating indefinite integrals because the derivative of a constant is zero. Therefore, when we integrate, we need to account for the possibility that there might have been a constant term in the original function. Now, we have the integral in terms of M, but we're not quite done yet. We need to substitute back our original expression for M to get the integral in terms of x.

Substituting Back for x

Okay, so we have M⁵/10 + C, and remember that M = x⁴ + x². Now we just substitute x⁴ + x² back in for M:

(x⁴ + x²)⁵ / 10 + C

And that's our final answer! We've successfully calculated the integral using substitution. Remember always to add that constant of integration, C, because that represents all functions with a derivative equal to the integrated function. By performing the substitution M = x⁴ + x², we transformed the original complex integral into a much simpler form that we could easily integrate. Then, by substituting back for x, we obtained the solution in terms of our original variable. This process highlights the power of substitution as a technique for simplifying integrals and making them more manageable. So there you have it, the final answer to the integral is (x⁴ + x²)⁵ / 10 + C.

Final Answer

So, to recap, the integral ∫(2x³ + x)(x⁴ + x²)⁴ dx is equal to (x⁴ + x²)⁵ / 10 + C. We found this by using the substitution M = x⁴ + x², which simplified the integral, allowing us to integrate it easily. Remember, when dealing with integrals, always look for ways to simplify the expression, whether through substitution, integration by parts, or other techniques. And don't forget to add the constant of integration, C, to account for all possible antiderivatives. So there you have it. Calculating integrals can be challenging, but with the right techniques and a bit of practice, you can master them. Keep practicing, and you'll become more comfortable with these concepts and able to tackle even more complex integrals. Keep up the great work! Remember that understanding these techniques not only helps with solving integrals but also builds a strong foundation for more advanced topics in calculus and beyond. Keep exploring and learning, and you'll continue to grow your mathematical skills.