Integral Of X 3 Sqrt 1 X 2

Solving the Integral of x³√(1-x²)

This article provides a comprehensive guide on how to solve the definite integral of x³√(1-x²). We will explore various methods, delve into the underlying mathematical concepts, and offer practical steps to arrive at the solution. This detailed explanation will be beneficial for students of calculus and anyone looking to strengthen their integration skills.

Understanding the Integral

The integral we aim to solve is:

∫x³√(1-x²) dx

This integral involves a composite function, making direct integration challenging. We'll explore the most effective approach: u-substitution.

Employing u-Substitution

U-substitution, also known as the reverse chain rule, is a powerful technique for simplifying integrals involving composite functions. The key is to choose a suitable substitution that simplifies the integrand.

Choosing the Right Substitution

In our integral, a strategic choice for substitution is:

u = 1 - x²

This choice simplifies the square root term. Now, we need to find the derivative of 'u' with respect to 'x':

du/dx = -2x

This allows us to express 'dx' in terms of 'du':

dx = -du/(2x)

Substituting and Simplifying

Substituting 'u' and 'dx' into our original integral, we get:

∫x³√u * (-du/(2x))

Notice that one 'x' cancels out, leaving:

-1/2 ∫x²√u du

We now need to express x² in terms of 'u'. Recall our substitution: u = 1 - x². Solving for x², we get:

x² = 1 - u

Substituting this back into our integral:

-1/2 ∫(1 - u)√u du

This integral is now significantly simpler and can be solved using standard integration techniques.

Solving the Simplified Integral

We can expand the integrand:

-1/2 ∫(√u - u√u) du

This can be rewritten using fractional exponents:

-1/2 ∫(u¹/² - u³/²) du

Now, we can apply the power rule of integration: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where 'C' is the constant of integration. Applying this rule to each term:

-1/2 [(u³/²) / (3/2) - (u⁵/²) / (5/2)] + C

Simplifying the fractions:

-1/2 [(2/3)u³/² - (2/5)u⁵/²] + C

Further simplification gives:

(-1/3)u³/² + (1/5)u⁵/² + C

Back-Substitution

Remember that our solution is in terms of 'u'. We need to substitute back 'u = 1 - x²' to express the solution in terms of 'x':

(-1/3)(1 - x²)³/² + (1/5)(1 - x²)⁵/² + C

The Final Solution

Therefore, the indefinite integral of x³√(1-x²) is:

(-1/3)(1 - x²)³/² + (1/5)(1 - x²)⁵/² + C

Verification through Differentiation

To verify our solution, we can differentiate the result and see if we get back the original integrand. This step is crucial for ensuring the accuracy of our integration. The differentiation process is quite involved and requires application of the chain rule multiple times. However, if done correctly, it will confirm that our integrated result is indeed correct.

Definite Integrals

If we were dealing with a definite integral, we would substitute the upper and lower limits of integration into the final solution and subtract the resulting values to obtain a numerical answer.

For instance, if we had the definite integral:

∫[from a to b] x³√(1-x²) dx

We would evaluate:

[(-1/3)(1 - b²)³/² + (1/5)(1 - b²)⁵/²] - [(-1/3)(1 - a²)³/² + (1/5)(1 - a²)⁵/²]

Exploring Alternative Methods

While u-substitution is the most efficient method for this particular integral, it's important to note that other techniques like trigonometric substitution might also be applicable, albeit potentially more complex in this specific case. Trigonometric substitution often proves useful when dealing with integrals containing expressions of the form √(a² - x²), √(a² + x²), or √(x² - a²).

Advanced Considerations

• Domain Restrictions: The original integrand, x³√(1-x²), has a restricted domain. The expression inside the square root (1-x²) must be non-negative, implying that -1 ≤ x ≤ 1. This restriction needs to be considered when evaluating definite integrals.

• Numerical Methods: For integrals that are difficult or impossible to solve analytically, numerical integration methods such as the trapezoidal rule or Simpson's rule can be employed to obtain approximate solutions.

• Applications: Integrals of this type appear in various fields, including physics and engineering, often related to problems involving volumes of revolution or calculations related to curved surfaces.

Conclusion

Solving the integral of x³√(1-x²) requires careful application of u-substitution and a methodical approach to simplifying the integrand. By following the steps outlined, you can successfully find the indefinite integral and apply the result to solve definite integrals. Remember to always verify your solution through differentiation and consider the domain restrictions of the original function. This detailed guide provides a solid foundation for tackling more complex integration problems in the future. The understanding gained from this example will strengthen your calculus skills and help you approach similar integral problems with confidence. The practice of verifying solutions through differentiation and understanding the underlying concepts will serve as valuable tools throughout your mathematical journey.