Calculating The Indefinite Integral Of X³√(5-x⁴) + C

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In the realm of calculus, indefinite integrals hold a prominent position, representing the family of functions whose derivatives all match a given function. Evaluating these integrals often involves a blend of algebraic manipulation, insightful substitutions, and a touch of creative problem-solving. In this article, we will embark on a step-by-step journey to calculate the indefinite integral of the function x³√(5-x⁴), making sure to include the crucial constant of integration, +C, which signifies the infinite possibilities within the antiderivative family.

The integral we aim to solve is:

∫ x³√(5-x⁴) dx

This integral might appear daunting at first glance, but with the strategic application of u-substitution, we can transform it into a more manageable form. U-substitution, a cornerstone technique in integral calculus, allows us to simplify complex integrals by replacing a portion of the integrand with a new variable, u, effectively reversing the chain rule of differentiation.

Step 1: Identifying the Appropriate Substitution

The heart of u-substitution lies in selecting the right expression to represent as u. A keen eye will notice that the derivative of 5-x⁴ is closely related to the x³ term present in the integral. This observation suggests that substituting u = 5-x⁴ might be a fruitful path to explore. By choosing this substitution, we aim to simplify the expression under the square root, paving the way for a smoother integration process.

Let's formally define our substitution:

u = 5 - x⁴

Step 2: Calculating the Differential du

With our substitution in place, the next step involves finding the differential du, which represents the derivative of u with respect to x, multiplied by dx. This differential will enable us to replace the x³ dx term in the original integral with an equivalent expression in terms of u and du.

Differentiating u = 5 - x⁴ with respect to x, we obtain:

du/dx = -4x³

Multiplying both sides by dx, we get:

du = -4x³ dx

Step 3: Adjusting the Integral for the Substitution

Now, we need to manipulate the original integral to perfectly match the du expression we just derived. Notice that our integral contains x³ dx, while du is equal to -4x³ dx. To bridge this gap, we can multiply both sides of the du equation by -1/4:

(-1/4) du = x³ dx

With this adjustment, we can now rewrite the original integral in terms of u and du. Replacing 5-x⁴ with u and x³ dx with (-1/4) du, we get:

∫ x³√(5-x⁴) dx = ∫ √u (-1/4) du

Step 4: Simplifying and Integrating

The integral is now significantly simpler. We can factor out the constant -1/4 from the integral:

∫ √u (-1/4) du = (-1/4) ∫ √u du

To further facilitate integration, let's express the square root of u as a power of u:

(-1/4) ∫ √u du = (-1/4) ∫ u¹/² du

Now, we can apply the power rule for integration, which states that the integral of uⁿ with respect to u is (uⁿ⁺¹)/(n+1), provided that n ≠ -1. In our case, n = 1/2, so we have:

(-1/4) ∫ u¹/² du = (-1/4) * (u^(1/2 + 1) / (1/2 + 1)) + C

Simplifying the exponent and the denominator, we get:

(-1/4) * (u^(3/2) / (3/2)) + C = (-1/4) * (2/3) * u^(3/2) + C

Multiplying the constants, we obtain:

(-1/4) * (2/3) * u^(3/2) + C = (-1/6) * u^(3/2) + C

Step 5: Back-Substituting for u

The final step in evaluating the indefinite integral is to substitute back the original expression for u in terms of x. Recall that we defined u = 5 - x⁴. Replacing u with this expression, we arrive at the final answer:

(-1/6) * u^(3/2) + C = (-1/6) * (5 - x⁴)^(3/2) + C

Therefore, the indefinite integral of x³√(5-x⁴) is:

∫ x³√(5-x⁴) dx = (-1/6) * (5 - x⁴)^(3/2) + C

Conclusion

Through the strategic application of u-substitution and the power rule for integration, we have successfully calculated the indefinite integral of x³√(5-x⁴). This process highlights the power and elegance of calculus techniques in tackling seemingly complex problems. Remember, the constant of integration, +C, is an indispensable part of the answer, acknowledging the infinite family of antiderivatives that differ only by a constant term. This comprehensive step-by-step solution not only provides the answer but also illuminates the underlying concepts and techniques involved in evaluating indefinite integrals. By mastering these techniques, students and enthusiasts can confidently navigate the world of calculus and unlock its vast potential for solving real-world problems.

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The final answer is (-1/6) * (5 - x⁴)^(3/2) + C