https://file.aiquickdraw.com/mathbot/file/1725811470066f8cphuz9.jpg

try to get the correct answer. 0 is not the answer!

Let's analyze the integral given in the image and work towards the correct answer.

### Problem Statement
We are asked to evaluate the integral:

\[
\int 8 \sin^3(x) \sqrt{\cos^2(x)} \, dx
\]

within the limits \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).

### Step 1: Simplify the Integrand
Notice that \( \sqrt{\cos^2(x)} = |\cos(x)| \). However, within the given limits, \( \cos(x) \) is non-negative, so \( \sqrt{\cos^2(x)} = \cos(x) \).

Thus, the integral simplifies to:

\[
\int 8 \sin^3(x) \cos(x) \, dx
\]

### Step 2: Use Substitution
Let \( u = \sin(x) \). Then, \( du = \cos(x) \, dx \).

The integral now becomes:

\[
\int 8 u^3 \, du
\]

### Step 3: Integrate
Integrating \( u^3 \):

\[
\int 8 u^3 \, du = 8 \cdot \frac{u^4}{4} = 2u^4 + C
\]

Since \( u = \sin(x) \), the integral becomes:

\[
2 \sin^4(x) + C
\]

### Step 4: Evaluate the Definite Integral
Now, evaluate this expression from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

However, because \(\sin(x)\) is symmetric around \(x = 0\), and we are integrating an even function over symmetric limits, the integral evaluates to zero, which is not expected based on the image.

### Step 5: Consider the Correct Evaluation
It appears the constant of integration should not be considered for definite integrals. Let's focus only on definite evaluation:

\[
\left[ 2 \sin^4(x) \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \sin^4\left(\frac{\pi}{2}\right) - \sin^4\left(-\frac{\pi}{2}\right)
\]

Given that \(\sin\left(\pm \frac{\pi}{2}\right) = 1\), both ends yield:

\[
1 - 1 = 0
\]

If there are no other terms involved, it will indeed result in zero, but since this is not the correct answer according to the problem, it means we likely need to revisit a different approach or integral setup.

### Step 6: Correct Substitution and Interpretation
Given the context, ensure you've applied substitution correctly, specifically:

- Checking the parity of sine in the symmetric interval.
- Considering any unique non-zero simplifications or conversions that could arise from complex cosine/sine identities or functions.

**In conclusion**: I’d recommend checking your original setup and see if there's a missing differential adjustment or identity that simplifies out what I interpreted as zero. The final setup might require a more nuanced trigonometric or substitution technique.

Evaluate the integral: ∫ 8 sin³(x) √(cos²(x)) dx for -π/2 ≤ x ≤ π/2. Note: Use an upper-case 'C' for the constant of integration.

Explore how to evaluate the integral ∫ 8 sin³(x) √(cos²(x)) dx over the range -π/2 ≤ x ≤ π/2. This problem demonstrates concepts in integration, trigonometric functions, and substitution methods, suitable for undergraduate calculus students.

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