**Using substitution:** Describe why \(\int x (5 - x^2)^3 \, dx \neq \int u^3 \, du\) where \(u = 5 - x^2\).### ExplanationTo understand why the integrals are not equivalent, we must consider the substitution process. The given substitution is \(u = 5 - x^2\). 1. **Differentiating**: By differentiating \(u\) with respect to \(x\), we get: \[ \frac{du}{dx} = -2x \quad \Rightarrow \quad du = -2x \, dx \quad \Rightarrow \quad x \, dx = -\frac{1}{2} du \]2. **Substitution in the Integral**: Substitute \(u\) and \(x \, dx\) in the original integral: \[ \int x (5 - x^2)^3 \, dx = \int (5 - x^2)^3 \cdot x \, dx = \int u^3 \cdot \left(-\frac{1}{2}\right) \, du = -\frac{1}{2} \int u^3 \, du \]This shows that the correct substitution results in a factor of \(-\frac{1}{2}\) in front of the integral of \(u^3\), clearly indicating that the expressions of these integrals are not directly equal without consideration of the factor introduced during substitution.

Name: **Using substitution:** Describe why \(\int x (5 - x^2)^3 \, dx \neq
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Using substitution:** Describe why \(\int x (5 - x^2)^3 \, dx \neq \int u^3 \, du\) where \(u = 5 - x^2\).### ExplanationTo understand why the integrals are not equivalent, we must consider the substitution process. The given substitution is \(u =