Problem 8 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int \sqrt[3]{\left(1-2 x^{2}\right)}(-4 x) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\sqrt[3]{1-2x^{2}}(-4x)\) dx is \(-\frac{3}{2}(1-2x^{2})^{2/3}\)

Step 1: Identify the Substitution

In the exercise, it can be observed that the derivative of \(1-2x^{2}\) is \(-4x\), which is present in the integrand. Therefore, it is reasonable to make the substitution \(u = 1-2x^{2}\). As a result, the differential \(du\) would be \(-4x dx\). Thus, replacing \(-4x dx\) with \(du\) simplifies the problem.

Step 2: Perform the Integration

After the substitution, the integral simplifies to \(-\int \sqrt[3]{u} du\). The antiderivative of this can be obtained directly through the power rule for integration, which states that the antiderivative of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), where \(n\) is any real number but -1. Thus, the antiderivative of \(-\sqrt[3]{u}\) is \(-\frac{3}{2}u^{2/3}\).

Step 3: Back-Substitute the Original Variable

Since the solution has to be in terms of the original variable \(x\), replace \(u\) with \(1-2x^{2}\) in the antiderivative to finally get the solution to the problem. This yields a solution of \(-\frac{3}{2}(1-2x^2)^{2/3}\).

Step 4: Check the Result by Differentiation

To confirm the answer, differentiate the result \(-\frac{3}{2}(1-2x^{2})^{2/3}\) with respect to \(x\) using the chain rule and ensure that the result matches the original integrand. That is, the derivative should be \(\sqrt[3]{1-2x^{2}}(-4x)\). This verifies the correctness of the integral.