Problem 42 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int 5 x \sqrt[4]{1-x^{2}} d x $$

Step-by-Step Solution

Answer

The integral is \(-2(1-x^2)^{5/4} + C\).

Step 1: Identify the integral form

We have the integral \( \int 5x \sqrt[4]{1-x^2} \, dx \). The expression inside the integral combines polynomial and root functions, and our goal is to simplify it or use substitution to find an antiderivative.

Step 2: Apply substitution

Let \( u = 1 - x^2 \). Then, \( du = -2x \, dx \), or equivalently, \( -\frac{1}{2} du = x \, dx \). Substitute into the integral: \( \int 5x \sqrt[4]{1-x^2} \, dx = \int 5 \sqrt[4]{u} \left(-\frac{1}{2} \right) du \). This simplifies to \( -\frac{5}{2} \int u^{1/4} \, du \).

Step 3: Evaluate the integral of a power

The integral \( \int u^{1/4} \, du \) can be evaluated using the formula for the integral of powers: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = \frac{1}{4} \), so \( \int u^{1/4} \, du = \frac{u^{5/4}}{5/4} + C = \frac{4}{5} u^{5/4} + C \).

Step 4: Substitute back

After integration, substitute \( u = 1 - x^2 \) back into the solution: \( -\frac{5}{2} \cdot \frac{4}{5} (1-x^2)^{5/4} + C = -2(1-x^2)^{5/4} + C \).

Step 5: Check by differentiating

Differentiate the result \( f(x) = -2(1-x^2)^{5/4} + C \) to verify the original integrand is obtained: \( f'(x) = -2 \cdot \frac{5}{4} (1-x^2)^{1/4} \cdot (-2x) \), using the chain rule. This simplifies to \( 5x(1-x^2)^{1/4} \), which matches the original integrand, \( 5x \sqrt[4]{1-x^2} \). The verification confirms the solution is correct.

Key Concepts

Antiderivative

In integral calculus, finding the antiderivative of a function is the reverse process of differentiation. Think of it as asking, "What function, when differentiated, will give me the function I'm starting with?"
In the given exercise, we're searching for an antiderivative, or a function whose derivative results back in the given integrand:

• Integrand: The expression under the integral sign.
• Prefix 'anti': Implying reverse or back.

Finding the antiderivative involves simplifying the expression or using known antiderivative formulas, a skill you'll develop over time. This exercise shows us an application of finding an antiderivative using a method called substitution.

Integration by Substitution

Integration by substitution is a technique used to simplify an integral by changing the variable. It's similar to the reverse of the chain rule in differentiation.
In this exercise, we set \( u = 1 - x^2 \), transforming the integral from one that seems complicated to one that is easier to integrate:

• Substitution: Replace a function or part of it with a single variable.
• Change of variable: Allows integration of more complex expressions.
• New Differential: Calculate \( du \) in terms of \( dx \).

After substitution, the integral becomes simpler. This exercise utilized this technique by substituting \( 1-x^2 \) with \( u \), which eventually simplified the process.

Differentiation

Differentiation is the process of finding the rate at which a function is changing at any given point. In this exercise, differentiation is used to verify that our antiderivative, when differentiated, gives back the integrand. This is an important check:

• By differentiating \( f(x) = -2(1-x^2)^{5/4} + C \), the goal was to return to the original integrand \( 5x \sqrt[4]{1-x^2} \).
• Chain Rule and Power Rule play a critical role in this verification.

This check helps ensure that everything was done correctly in the integration process by confirming that the derivative of the found solution returns to the initialized problem.

Integral of Powers

Understanding how to integrate powers is crucial in finding solutions to many integrals, especially those that arise in substitution:

• The formula \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) is fundamental when dealing with powers, except when \( n = -1 \).
• In this problem, \( n = \frac{1}{4} \), so substituting into this formula gives us the antiderivative.

Recognizing powers and properly using this simple integration rule will help you solve many integrals efficiently, just like simplifying \(- \frac{5}{2} \cdot \frac{4}{5} u^{5/4}\), resulting in the final solution: a cleaned-up expression that we then transform back into \( x \).