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The problem is asking to compute the following definite integral:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4(x) \, dx
\]

We will solve this integral step by step.

### Step 1: Express \(\sin^4(x)\) in terms of simpler trigonometric functions
Using the power reduction identity for sine:

\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]

we can write:

\[
\sin^4(x) = \left(\sin^2(x)\right)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2
\]

Expanding this:

\[
\sin^4(x) = \frac{1}{4}(1 - 2\cos(2x) + \cos^2(2x))
\]

### Step 2: Simplify further using the identity for \(\cos^2(2x)\)
We use the identity:

\[
\cos^2(2x) = \frac{1 + \cos(4x)}{2}
\]

Substitute this into the expression for \(\sin^4(x)\):

\[
\sin^4(x) = \frac{1}{4} \left( 1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2} \right)
\]

Simplifying:

\[
\sin^4(x) = \frac{1}{4} \left( \frac{3}{2} - 2\cos(2x) + \frac{\cos(4x)}{2} \right)
\]

### Step 3: Break the integral into parts
Now, integrate each part from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\):

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4(x) \, dx = \frac{1}{4} \left[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{3}{2} \, dx - 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(2x) \, dx + \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(4x) \, dx \right]
\]

#### First part:
\[
\frac{1}{4} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{3}{2} \, dx = \frac{1}{4} \times \frac{3}{2} \times \pi = \frac{3\pi}{8}
\]

#### Second part:
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(2x) \, dx = 0 \quad (\text{since cosine is an odd function and symmetric about the origin})
\]

#### Third part:
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(4x) \, dx = 0 \quad (\text{same reasoning as above})
\]

### Step 4: Final result
Thus, the value of the integral is:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4(x) \, dx = \frac{3\pi}{8}
\]

The correct answer is option **(b)** \(\frac{3\pi}{8}\).

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. What is the integral of \(\sin^2(x)\) over the same limits?
2. How can the power reduction identities be derived?
3. What is the integral of \(\sin^6(x)\) over \([-\frac{\pi}{2}, \frac{\pi}{2}]\)?
4. How does symmetry of functions affect definite integrals?
5. How can we apply substitution to solve similar integrals involving powers of trigonometric functions?

**Tip**: Using symmetry properties of trigonometric functions can simplify complex integrals significantly.

The value of the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4(x) dx\)

Calculate the definite integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4(x) dx\) using power reduction identities. This step-by-step guide simplifies the process of solving integrals of higher powers of sine functions.

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